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JUl. 



THEORETICAL MECHANICS. 



MECHANICS OF ENGINEERING. 



THEORETICAL MECHANICS, 



WITH AN 



INTRODUCTION TO THE CALCULUS. 



DESIGNED AS A TEXT-BOOK 

FOR TECHNICAL SCHOOLS AND COLLEGES, AND FOR THE USE 
OF ENGINEERS, ARCHITECTS, ETC. 



BY 

JULIUS WEISBACH, Ph.D, 

M 
OBERBEBGBATH AND PROFESSOR AT THE ROYAL MINING ACADEMY AT FREI- 
BERG ; MEMBER OF THE IMPERIAL ACADEMY OF SCIENCES AT ST. 
PETERSBURG, ETC. 



Translated from the Fourth Augmented and Improved German Edition by 

ECKLEY B. COXE, A.M., 

MINING ENGINEER. 



SEVENTH AMERICAN EDITION. 
rLLUSTRATED WITH 903 WOOD-CUTS IN THE TEXT. 



New York : 

D. VAN NOSTRAND, PUBLISHER, 

23 Murray and 27 Warren Street. 

1885. 






Entered, according to Act of Congress, in the year 1875, by 

D, VAN" NOSTRAND, 

In the office of the Librarian of Congress, at Washington, D, G, 






L^dC Aj; 



PREFACE TO THE FIRST EDITION. 



XT is not without apprehension that I give to the pubhc my 
elementary treatise upon the Mechanics of Engineering and 
of the Construction of Machines. Although I can say to myself 
that, in preparing this manual, I have gone to work with all pos- 
sible care and attention, yet I fear that I have not been able to 
satisfy the wishes of every one. The ideas, wishes and require- 
ments of the public a,re so various, that it is not possible to do 
so. Some may find the treatment of a particular subject too 
detailed, others perhaps too short ; some will desire a more 
scientific discussion of certain subjects, while others would prefer 
one more popular. Many years of study, much experience in 
teaching and very varied observations and experiments have led 
me to adopt, as most suitable to the object in view, the method, 
according to which this work has been arranged. My principal 
effort has been to obtain the greatest simplicity in enunciation 
and demonstration, and to treat all the important laws, in their 
practical applications, without the aid of the higher mathematics. 
If we consider how many subjects a technical man must master in 
order to accomplish any thing very important in his profession, 
we must make it our business as teachers and authors for techni- 
cal men to facihtate the thorough study of science by simplicity 
of diction, by removing whatever may be unnecessary, and by em- 
ploying the best known and most practicable methods. For this 
reason I have entirely avoided the use of the Calculus in this 
work. Although at the present time the opportunities for ac- 
quiring a knowledge of it are no longer rare, yet it is an unde- 
niable fact that, unless we are constantly making use of it, we 
soon lose that facility of calculation, which is indispensable ; for 
this reason so many able engineers can no longer employ the Cal- 



vi PREFACE TO THE FIRST EDITION. 

cuius whicli they learned in their youth. As I do not agree with 
those authors, who in popular treatises enunciate without proof 
the more difficult laws, I have preferred to deduce or demon- 
strate them in an elementary, although sometimes in a somewhat 
roundabout, manner. 

Formulas without proof will therefore seldom be found in this 
work. We will assume that the reader has a general knowledge 
of certain principles of natural philosophy and a thorough knowl- 
edge of the elements of pure mathematics. My attention has 
been especially directed to preserving the proper mean between 
generalization and specialization. Although I appreciate the ad- 
vantages of generalization, yet it is my opinion that in this work, 
as in all elementary treatises, too much generalization is. to be 
avoided. The simple is oftener met with in practice than the 
complex. It is also undeniable that in considering the general 
case we often fail to attain a more profound knowledge of the 
special one, and that it is often easier to deduce the complex from 
the simple than the simple from the complex. The reader must 
not expect to find in this work a treatise upon the construction 
of machines, but only an introduction to or preparation for it. 
Mechanics should bear the same relation to the construction of 
machines that Descriptive Geometry does to Mechanical Drawing. 
When the pupil has acquired sufficient knowledge of Mechanics 
and of Descriptive Geometry, it appears better to combine the 
course of Construction of Machines with that of Mechanical 
Drawing. 

It may be doubted whether it was advisable to divide my sub- 
ject into two parts, theoretical and applied. If we remember that 
this work is intended to give instruction upon all the mechanical 
relations of the construction and of the theory of machines, the 
advantage, or rather, the necessity, of such a division becomes 
evident. In order to judge of a structure or of a machine, we 
must have a knowledge of mechanical principles of a very varied 
character, e.g., those of friction, strength, inertia, impact, efflux, 
&c. ; the material for the mechanical study of a structure or of a 
machine must, therefore, be gathered from almost all the divis- 
ions of mechanics. Now, since it is better to study all the me- 
chanical principles of a machine at once than to collect them from 
all the different parts of mechanics, the advantage of such a di- 
vision is apparent. » 

Having practical application always in view, I have endeav- 
ored, in preparing my work, to illustrate the principles laid down 



PREFACE TO THE FIRST EDITION. vii 

in it by examples taken from every-day life. I am justified in 
asserting that this work contrasts favorably with any other of the 
same character in the number of appropriate examples, which are 
solved in it. I also hope that the great number of carefully-pre- 
pared figures will contribute to the object in view. My thanks 
are due to the publishers for having given the book in all respects 
the best appearance. Particular care has been taken to have the 
calculations correct ; generally every example has been calculated 
three times, and not by the same person. It is, therefore, im- 
prol^able that any gross errors will be found in them. In the ex- 
amples, as in the formulas, I have employed the Prussian weights 
and measures, as they are probably familiar to the majority of ni}^ 
readers. The printing (in this case so difficult) is open to little 
complaint. The mistakes in copying, or of impression, which 
have been observed, are noted at the end of the book, 

I do not think that many additions to this list need be made. 
An attentive examination of the illustrations will show that they 
have been prepared with care. The larger illustrations, particu- 
larly those representing bodies in three dimensions, are drawn 
according to the method of Axonometric Projection, first treated by 
me (see Polytechn. Mittheilungen Band I. Tubingen, 1845). 
This method of drawing possesses all the advantages of Isometric 
Projection, while in addition the pictures, which it furnishes, are 
not only more beautiful in themselves, but more easily awaken in 
us distinct conceptions of the objects represented. The drawings 
in this vv^ork are made in such a way that the dimensions of the 
width or depth appear but one-half as large as those of the height 
and length of the same size. I cannot omit thanking Mr. Ernest 
Roting, student at the academy in Freiberg, whose revision has 
essentially contributed to the accuracy of the work 

It is necessary to inform the reader that he will find much 
new matter, which is peculiar to the author. "Without stopping to 
mention many small articles, which occur in almost every chapter, 
I would call attention to the following comprehensive discussions : 
A general and easy determination of the centre of gravity of plane 
surfaces and of polyhedra, limited by plane surfaces, will be found 
in paragraphs 107, 112, and 113 ; an approximate formula for the 
catenary in paragraph 148 ; additional remarks upon the friction 
of axles in paragraphs 167, 168, 169, 172, and 173. Important 
additions to the theory of impact have been made, particularly 
in paragraphs 277 and 278 ; for heretofore the impact of imper- 
fectly elastic bodies has been too little considered, and that of a 



mi PREFACE TO THE FIRST EDITION. 

perfectly elastic with an imperfectly elastic body has not been 
treated at all. Very important additions, and in some cases en- 
tirely new laws, will be found in the chapter upon hydraulics, a 
subject to which I have for a number of years devoted special 
study. The laws of incomplete contraction, first observed by the 
author, will be found for the first time in a manual of mechanics. 
The author has also incorporated in it the principal results, so 
important in practice, of his experiments upon the efflux of water 
through oblique short pipes, elbows, curved and long pipes, etc., 
although the third number of his " Untersuchungen im Gebiete 
der Mechanik und Hydraulik " has not yet appeared. The chap- 
ter upon running water, upon hydrometry and upon the impact 
of water contains some original matter* The theories of the re- 
action of water discharging from a vessel and of the impact of 
water, which are treated according to the principle of mechanical 
effect, are original. 

I cannot, however, conceal from the reader that, since the vol- 
ume has been finished, I have wished that some few subjects had 
been treated differently ; but I must add that as yet I have ob- 
served no great imperfections. If at times the reader should 
miss something, he is referred to the second volume, which will 
supply both the accidental and the intentional omissions, as has 
been noted in many places in the first volume. 

The printmg of the second volume will now go on without in- 
terruption, so that we may expect the complete work to be in the 
hands of the reader before the end of the year. The pocket-book, 
the " Ingenieur," cited in the Mechanics, which contains a collec- 
tion of formulas, rules and tables of arithmetic, geometry and 
mechanics, will soon appear. 

It will be a source of great pleasure and satisfaction to me, if 
I have accomplished the purposes for which this work has been 
undertaken, namely, to give to the practical man a useful coun- 
sellor in questions of application, to the teacher of practical 
mechanics a serviceable text-book for instruction, and to the stu- 
dent of engineering a welcome aid in the study of mechanics. 

JULIUS WEISBACH. 

Freiberg, March lUh, 1846. 



PREFACE TO THE SECOND EDITION. 



^r^HE present (second) edition of the Mechanics of Engineering 
and of the Construction of Machines has undergone no es- 
sential alterations either in method or arrangement. The inter- 
nal construction of the work has been changed in many places, 
and its size has been considerably increased. The author has 
also endeavored, as much as possible, to correct the errors and 
omissions of the first edition. The great increase in size is 
mainly due to three additiojis. The first consists of a condensed 
Introduction to the Calculus, which has been made as popular as 
possible, and has been prefixed to the main work. The object of 
introducing it was to avoid too complicated and too artificial de- 
monstrations by means of the lower mathematics, and also to 
render the reader more independent in his study of mechanics, 
and to place him upon a higher stand-point in this important 
branch of science. By making use of the principles explained in 
the Introduction, it was possible to discuss many subjects of great 
practical importance, which previously we could not treat at all, 
or, at least, only imperfectly with the aid of elementary algebra 
and geometry. In order to avoid interruptions to those who 
have not made themselves familiar with the Elements of the Cal- 
culus, prefixed to the work, all the paragraphs, in which it is ap- 
plied, are designated by a parenthesis ( ). 

The second addition consists of a new chapter on Hydrostatics, 
in which the molecular action of water is treated. Since a knowl- 
edge of the molecular forces (capillarity) is of importance in ex- 
periments and observations in hydraulics and pneumatics, the 
author has thought it advisable to treat the fundamental princi- 
ples of these forces in a separate chapter. Finally, a chapter has 
been added to the work in the form of an appendix, which treats 



X PREFACE TO THE SECOND EDITION. 

of oscillation and wave motion. The author found himself com- 
pelled to do this in consequence of the importance to the engineer 
of a more accurate knowledge of the theory of oscillation. The 
great influence of vibration upon the working and durabihty of 
machines is a subject to which too much attention cannot be 
given. It is also to observations of oscillations that we owe the 
latest determination of the modulus of elasticity, which is of such 
importance in practice. I have mentioned in the Appendix the 
magnetic force, principally because it is of great use to the engi- 
neer in determining directions in mines, where the access to day- 
light is not easy. The theory of water-waves, which closes the 
volume, is a part of hydraulics ; its presence in this work requires, 
therefore, no explanation. Unfortunately, it is far from complete. 
The changes in the other parts of the work are the following : 
the chapter upon elasticity and strength has been much extended 
and altered, the subject of hydi-aulics has been treated more at 
length, and some modifications in it have been made, in conse- 
quence of the continued experiments of the author. 

I trust that the present edition will be received with the same 
favor as the last, by which the author was encouraged to continue 
his preparation of the work. 

JULIUS WEISBACHo 

Fbeibebg, May IhtTi, 1850. 



PREFACE TO THE THIRD EDITION. 



rriHE third edition of the first volume of my Mechanics of En- 
gineering and of the Construction of Machines, which I now 
give to the pubhc, has, compared with its predecessors, not only 
been improved, but also augmented and completed. The changes 
are due principally to the advance of science, and in some cases 
to the results of more recent investigations. When not withheld 
by some good reason, I have endeavored, so far as possibly, to 
satisfy the wishes which have been communicated to me from 
different quarters in regard to the work. From the extraordi- 
narv favor, with which it has been received both in and out of 
Germany, on this as well as on the other side of the Atlantic, I 
flatter myself that it has suited both in method and size the 
greater portion of the public for whom it was intended, and my 
efforts in preparing the new edition have been naturally directed to 
removing any errors or omissions, that have been observed, and 
to incorporating in it the latest experiments, treated in the same 
manner and as concisely as possible. I am sorry to be obliged to 
remark that the work has been subjected to unjust criticism. 
Thus, E.G., Professor Wiebe, of Berlin, in a remark upon pages 
245 and 246 of his work upon " die Lehre von der Befestigung 
der Machinentheile," (Berlin, 1854), states that I have given 
coefficients of torsion for square shafts in my Mechanics (first 
edition), as well as in the "Ingenieur," 16 times gi-eater than 
those given by Morin. The Professor has here committed an 
oversight ; for in my formulas, as is expressly stated in both 
works, the foui^th power of the half length of the side occurs, 
while the formulas of Morin and Wiebe, as well as those of my 
second edition, contain the fom'th power of the whole length of 
the side of the cross-section. Now since 2* is equal to 16, the 



xii PREFACE TO THE THIRD EDITION. 

error observed bj Professor Wiebe proceeds from a mistake on 
his part. 

I shall make no reply to the partial criticism contained in 
Grunerfs Archtv der Mathematik, as I do not wish to enter upon 
a useless controversy here. Besides, Professor Grunert has 
already j^rinted in his Archiv enough nonsense about Physics 
and Practical Mechanics (as I can easily prove) to demonstrate 
his unfitness for criticising works on those subjects. 

It would have been easier for me to have given my book a 
more scientific form ; but it would then have met with less favor, 
as it is intended for practical men. 

From another stand-point also the book can easily and with 
equal injustice be found fault with. Any one, who has had some 
practical experience, will have observed how little theory is made 
use of, and how often it is put in the back-ground and looked 
upon with disfavor by practical men. The fault of this is no 
doubt due in great measure to that method of instruction, which 
condemns the study of science for the sake of its applications. 

This edition, which has been augmented principally by the 
revision of the theory of elasticity and strength, and by the in- 
troduction of the latest hydraulic experiments, excels its prede- 
cessors not only in substance, but also in appearance, all the 
illustrations being new. The printing of the second volume will 
continue uninterruptedly. 

JULIUS WEISBACHo 

Freiberg, Jul^, 1856. 



PREFACE TO THE FOURTH EDITION. 



nnHE fourth edition of my Mechanics of Engineering and of 
the Construction of Machines has undergone no change eithet 
in method or arrangement. As three large editions have been ex- 
hausted in a comparatively short time, as two have been published 
in the English language, one in England and one in North Amer- 
ica, and as the work has been translated into Swedish, Polish, and 
Russian, I may well hope that this manual has met the wishes and 
needs of that great practical public fcr whom it is intended. I 
have, therefore, in preparing this edition, endeavored simply to 
remove any errors or omissions, which may have been observed, 
and to introduce the results of the latest practically important 
experiments, together with the newest developments of theory. 
Thus, E.G., in the chapter upon friction I have included the results 
of the latest experiments by Bochet, and the section upon elasti- 
city and strength has been rewritten in accordance with the 
present stand-point of science, in doing which I have made use of 
the recent works of Lame, Rankine, Bresse, etc. The section 
upon hydraulics has bsen augmented, improved and completed. 
The later researches of the author are here discussed. I will men- 
tion more particularly the experiments upon the efflux of water 
under great and very great pressures, as well as upon the heights 
of jets, those upon the efflux of air, and the comparative experi- 
ments upon the impact of streams of air and water. The chapter 
upon the efflux of air has been entirely rewritten, as the author is 
of the opinion that the ordinary formulas for the efflux of air 
under high pressures do not represent the law of efflux. The 
formulas obtained are very simple, since, without materially affect- 
ing its accuracy, I have substituted in the well-known formula 
for heat 



xiv PREFACE TO THE FOURTH EDITION. 

1 + (5t , ^ / y,x°." 
I + 6t \y) 
0,50 instead of the exponent 0,42, by which I obtain 

I^ = l/| (- § 461). 

The practical value of a formula does not depend upon its cor- 
rectness even at extreme limits, but rather upon the fact that, 
within given limits, it furnishes values which agree sufficiently 
well with the results of experiment. 

Several new paragraphs, in which Phoronomics and Aerosta- 
tics are treated with the aid of the Calculus, have been added. In 
hydraulics the pressure of water flowing through pipes, on account 
of its practical importance, has been treated separately in two new 
paragraphs (§ 439 and § 440). In the chapter upon the force and 
resistance of water I have treated the theory of the simple reaction 
wheel, as well as its application as an instrument for proving the 
theory of the impact and resistance of water. The more recent 
gas and water meters are also discussed, since these instruments 
are set in motion by the reaction of the issuing fluid, the intensity 
of which can easily be determined by the foregoing theory. 

Finally, the Appendix has been slightly augmented by the in- 
troduction of the report of the recent experiments of Geh. Ober- 
baurath Hagen upon, waves of water. 

In answer to the criticism, which has been made in some 
quarters, that a more scientific treatment of the subject, based 
upon the Calculus, would have been more in accordance with the 
object of the book, I would state that my book is intended for 
the use of practical men, who often do not possess either the 
requisite knowledge of the Calculus or sufficient facility in the use 
of it. Having labored during upwards of thirty years as instructor 
in a technical institution, during which time I have been engaged 
in practical works of various kinds and have made many journeys 
for the purpose of technical studies, I can confidently give an 
opinion upon this subject. 

As I consider my reputation as an author of much more 
importance than any mere pecuniary advantage, it is always a 
pleasure to me to find my " Mechanics " made use of in works of 
a similar character ; but when writers avail themselves of it with- 
out the slightest acknowledgment, I can only appeal to the judg- 
ment of the public. 

JULIUS WEISBACH. 

Fkeiberq, Map, 1863. 



TRANSLATOR'S PREFACE. 



rpHE favor, with which both the English and American editions 
of the Mechanics of Engineering and of the Construction of Ma- 
chines were received, would sufficiently justify the appearance of a 
new one, even if the original worS: had undergone no change. But 
as the first two volumes of the last (fourth) G-erman edition contain 
more than twice as much matter as those of the first, and since a 
third volume of about fifteen hundred pages has been added, th& 
translator feels not only that the work may be considered a new 
one, but also that, in offering it to the public, he is supplying a. 
real want. The text of this edition has been, to a great extent, 
rewritten and rearranged, and the translation is entirely original. 

Weisbach^s Mechanics is now so well known, wherever that sci- 
ence is taught, that any eulogy on our part would be superfluous. 

A large number of typographical errors, observed in the German 
edition, have been corrected with the approbation of the author, 
who has also communicated to the translator some slight modifica- 
tions in the text. The work of translation was begun with the 
author's approval, while the translator was a student of the Mining 
Academy at Freiberg, but the work was delayed by his professional 
engagements. He hopes that it will now appear without interrup- 
tion. 

At the suggestion of the author, an Appendix has been added 
containing an account of the articles upon the subjects treated in 
this volume, which have been published by him since the appear- 
ance of the last German edition. 



^"n 



TRANSLATOR'S PREFACE. 



All the tables, formulas, examples, etc., in which the Prussian 
weights and measures occur, haye been transformed so as to be ap- 
plicable to the EngUsh system. "Where the metrical system was 
employed in the original work, it has been retained in the transla- 
tion, as the meter is now much used both in England and America. 

The " Ingenieur," which is so often quoted in this work, has, 
unfortunately, not been translated into English, but all the refer- 
ences to it have been preserved, as the work is a valuable one, even 
to those who have little or iio knowledge" of German, and perhaps 
an English edition of it may be published. 

A list of errors and omissions observed in this volume will be 
given in the succeeding one, and the translator will be glad to be 
informed of any typographical errors. 

He would call attention to the illustrations, which are printed 
from electrotype copies of the wood-cuts prepared for the German 
edition, and his thanks are due to the publisher and stereotypers 
for the excellent appearance of the work. 

EOKLEY B. COXE. 



CONTENTS. 



INTRODUCTION TO THE CALCULUS. 

ABTICLE PAaa 

1 — 4 Functions. Laws of Nature 33 

5 — 6 Differential. Position of tangent 38 

7 — 8 Rules for differentiating 40 

9—10 The function y = x"^ 44 

11 — 12 Straight line, ellipse, hyperbola 49 

18 — 14 Course of curves, maximum and minimum 53 

15 McLaurin's series, binomial series 57 

16 — 18 Integral, Integral Calculus 60 

19 — 23 Exponential and logarithmic functions 63 

24 — 27 Trigonometrical and circular functions 70 

28 Integration by parts 76 

29 — 31 Quadrature of curves 78 

32 Rectification of curves 85 

38 — 34 Normal and radius of curvature 87 

35 Function y= -^ 93 

36 Method of the least squares 95 

37 Method of interpolation 98 



SECTION I. 

PHORONOMICS, OR THE PURELY MATHEMATICAL THEORY OF 

MOTION. 

CHAPTEK I. 

SIMPLE MOTIOlj". 

§ 1 Rest and motion. 105 

2— 3 Kinds of motion 105' 

4 — 6 Uniform motion 106 

7 — 9 Uniformly variable motion 107 

10 — 13 Uniformly accelerated motion 109 

14 Uniformly retarded motion 112 

£> 



xviii TABLE OF CONTENTS. 

PAGE 

§ 15 — 18 The free fall and vertical ascension of bodies 113 

19 Variable motion in general .' 117 

30 Difierential and integral formulas of phoronomics 119 

21 Mean velocity 121 

22 — 26 Graphical representation of the formulas of motion 122 

CHAPTEE II. 

COMPOUND MOTIOiq". 

27 — 29 Composition of motions 126 

30 Parallelogram of motions 127 

31 — 33 Parallelogram of velocities 128 

34 Composition and decomposition of velocities 131 

35 Composition of accelerations 132 

36 Composition of velocities and accelerations 132 

37 — 38 Parabolic motion 134 

39 Motion of projectiles 136 

40 Jets of water 138 

41^43 Curvilinear motion in general 141 

44 Application of the Calculus 145 

45 — 46 Relative motion 149 



SECTION II. 

MECHANICS, OR THE PHYSICAL SCIENCE OP MOTION IN 

GENERAL. 

CHAPTER I. 

FUNDAMENTAL PEINCIPLES AND LAWS OF MECHANICS. 

47 Mechanics, phoronomics, cinematics 154 

48 Force, gravity - 154 

49 Equilibrium, statics, dynamics 155 

50 Classification of the forces, motive forces, resistances, etc 155 

51 — 52 Pressure, traction, equality of forces 156 

53 Matter, material bodies. .„ 156 

54 Unit of weight, gram, pound 157 

55 Inertia 157 

56 Measure of forces 158 

57 — 59 Mass, heaviness 158 

60 — 61 Specific gravity, table of specific gravities 161 

62 State of aggregation 1^2 

63 Classification of the forces 1^^ 

64 Forces, how determined. . , 163 

65 Action and reaction 164 

66 Division of mechanics. 164 



TABLE OF COISTTEISTTS. xix 

CHAPTEK II. 

MECHANICS OF A MATERIAL POINT. 

PAGE 

§ 67 Material point 165 

68 — 69 Simple constant force 166 

70 — 73 Mechanical effect or work done bj a force 168 

74 — 75 Principle of tlie vis viva 171 

76 Composition of forces 174 

77 Parallelogram of forces 177 

78 Decomposition of forces 179 

79 — 80 Forces in a plane 180 

81 Forces in space 182 

83 — 88 Principle of virtual velocities 185 

84 Transmission of mechanical effect 187 

85 Work done in curvilinear motion 189. 

SECTION III. 

STATICS OF RIGID BODIES. 
CHAPTER I. 

GEIOIRAL PRINCIPLES OF THE STATICS OF RIGID BODIES. 

86 — 87 Transference of the point of application 192 

88—89 Statical moment , 193 

90 — 91 Composition of forces in the same plane. 195 

92 Parallel forces 199 

93—95 Couples 200 

96 Centre of parallel forces 205 

97 Forces in space 207 

98 — 102 Principle of virtual velocities 209 

CHAPTER II. 

THE THEORY OF THE CENTRE OF GRAVITY. 

103 — 104 Centre of gravity, line of gravity, plane of 'gravity 218 

105 — 106 Determination of the centre of gravity 214 

107—108 Centre of gravity of lines 216 

109 — 114 Centre of gravity of plane figures 218 

115 Determination of the centre of gravity by the Calculus 226 

116 The centre of gravity of curved surfaces 227 

117—123 Centre of gravity of bodies 228 

124 Applications of Simpson's rule 237 

125 De'tcrmination of the centre of gravity of solids of rotation, etc. , 239 
126—128 Properties of Guldinus 241 



XX TABLE OF CONTENTS. 

CHAPTEE III. 

EQUILIBRIUM OF BODIES EIGIDLY FASTENED AK"D SUPPORTED. 

PAGE 

§ 129 Method of fastening 24:7 

130 Equilibrium of supported bodies 248 

131 Stability of a suspended body 349 

182 — 138 Pressure upon the points of support of a body 250 

184 Equilibrium of forces around an axis 254 

185 — 187 Lever, mathematical and material 255 

138 — 139 Pressure of bodies upon one another 261 

140—141 Stability 263 

142 — 143 Formulas for stability 266 

144 Dynamical stability 269 

145 Work done in moving a heavy body 271 

146 Stability of a body upon an inclined plane 272 

147 Theory of the inclined plane 274 

148 Application of the principle of virtual velocities 275 

149 Theory of the wedge 27T 

OHAPTEK IV. 

EQUILIBRIUM IK FUNICULAR MACHINES. 

150 Funicular machines, funicular polygon 280 

151 — 153 Fixed and movable knots 281 

154 — 156 Equilibrium of a funicular polygon 286 

157 The parabola as catenary 291 

158—160 The catenary 293 

161—162 Equation of the catenary 299 

163—164 The pulley, fixed and movable 303 

165 — 166 Wheel and axle, equilibrium of the same 305 

CHAPTEE Y. 

THE RESISTANCE OF FRICTION AND THE RIGIDITY OF CORDAGE. 

167—168 Friction 809 

169 Kinds of friction, sliding and rolling 810 

170 Laws of friction 311 

171 Coefficient of friction 313 

172 Angle of friction and cone of friction 814 

178 Experiments on friction 815 

174 Friction tables 818 

175 Latest experiments on friction 320 

176—177 Inclined plane, friction upon an inclined plane 323 

178 The theory of the equilibrium of bodies with reference to the 

friction 328 

179—180 Wedge, friction on the wedge 329' 

181 — 185 Coefficients of friction of axles, friction of axles 33S 



TABLE OF CONTENTS. xxi 

PAGE 

§ 186 Poneelet's theorem. 341 

187 Lever, axial friction of the lever 343 

188 Friction of a pivot 345 

189 Friction on conical pivots 347 

190 Anti-friction pivots 349 

191 Friction on points and knife-edges 352 

193 RolUng friction 353 

193 — 194 Friction of cords and chains 356 

195 Rigidity of chains 361 

196—200 Rigidity of cordage 363 

SECTION IV. 

THE APPLICATION OF STATICS TO THE ELASTICITY AND 
STRENGTH OF BODIES. 

CHAPTER I. 

ELASTICITY Al^D STKEI^GTH OF EXTENSION", COMPEESSION 

AND SHEAKING. 

201 Elasticity of rigid bodies 371 

202 Elasticity and strength 372 

203 Extension and compression 374 

204 Modulus of elasticity 376 

205 Modulus of proof strength, modulus of ultimate strength 379 

206 Modulus of resilience and fragility 382 

207 Extension of a body by its own weight 384 

208 Bodies of uniform strength 387 

209 Experiments upon extension and compression 391 

210 Experiments upon extension . . . « 393 

211 Elasticity and strength of iron and wood 397 

212 Numbers determined by experiment 401 

213 Strength of shearing 406 

CHAPTER II. 

ELASTICITY AND STEENGTH OF FLEXUEE OE BENDING. 

214 Flexure of a rigid body 409 

215 Moment of flexure ( TT) 412 

216—217 Elastic curve 414 

218 More general equation of the elastic curve 419 

219—222 Flexure produced by two parallel forces 422 

223 A uniformly loaded girder 430 

224 — 225 Reduction of the moment of flexure 432 

226 Moment of flexure of a strip 435 

227 Moment of flexure of a parallelopipedical girder 436 

228 Hollow, double-webbed or tubular girders 437 



xxii TABLE OF C0KTENT8. 

PA&E 

§ 229 Triangular girders 439 

230 Polygonal girders 441 

231 Cylindrical or elliptical girders 443 

232 Application of the calculus to tlie determination of W 445 

233 — 234 Beams with curvilinear cross-sections 447 

235 Strength of flexure 450 

236 Formulas for the strength of bodies 453 

237 Difference in the moduli of proof strength 457 

238 Difference in the moduli of ultimate strength 460 

239 Experiments upon flexure and rupture. 463 

240 Moduli of proof and ultimate strength 466 

241 Relative deflection 469 

242 Moments of proof load 472 

243 Cross-section of w^ooden girders. 474 

244 Hollow and webbed girders 477 

245 Eccentric loads : 480 

246 — 248 Girders supported in different ways 484 

249 — 250 Girders not uniformly loaded 491 

251 — 252 Cross-section of rupture 494 

253 — 254 Bodies of uniform strength 498 

255 Flexure of bodies of uniform strength 504 

256 Deflection of metal springs 506 

CHAPTEE III. 

THE ACTION" OF THE SHEAKING ELASTICITY I]!T THE BEN"DIN"G 

AND TWISTING OF BODIES. 

257 The shearing force parallel to the neutral axis. 510 

258 The shearing force in the plane of the cross-section 513 

259 Maximum and minimum strain , .... 515 

260 Influence of the strength of shearing upon the proof load of a 

girder 519 

261 Influence of the elasticity of shearing upon the form of the 

elastic curve 522 

262 Elasticity of torsion 523 

263 Moment of torsion or twisting moment 524 

'. 264 Resistance to rupture by torsion 528 

CHAPTEE IV. 

ON THE PROOF STRENGTH OF LONG COLUMNS, OR THE RESIST- 
ANCE TO CRUSHING BY BENDING OR BREAKING ACROSS. 

S65— 266 Flexure and proof load of long pillars 532 

267 Bodies of uniform resistance to breaking across 539 

268 Hodgkinson experiments 542 

269 More simple determination of the proof load 544 



TABLE OF CONTENTS. xxiii 

CHAPTER V. 

COMBIITED ELASTICITY AKD STEENGTH. 

PAea 

§ 270 Combined elasticity and strength 547 

271 Eccentric pull and thrust 551 

272—273 Oblique pull or thrust 553 

274 — 275 Flexure of girders subjected to a tensile force 559 

276 Torsion combined with a tensile or compressive force 563 

277 Flexure and torsion combined 567 

278 Bending forces in different planes • . 570 



SECTION V. 

DYNAMICS OF RIGID BODIES. 
CHAPTER I. 

THE THEORY OF THE MOMEKT OF I2!^ERTIA. 

279 Kinds of motion 573 

280 Rectilinear motion • • 574 

281 Motion of rotation 575 

282 Moment of inertia 576 

283 Reduction of the mass 578 

284 Reduction of the moments of inertia 580 

285 Radius of gyration 581 

286 Moment of inertia of a rod 582 

287 Rectangle and Parallelopipedon (moments of inertia of) 583 

288 Prism and Cylinder 585 

289 Cone and Pyramid 587 

290 Sphere 588 

291 Cylinder and Cone 589 

292 Segments 590 

293 Parabola and Ellipse 592 

294 Solids and surfaces of revolution 593 

295 — 296 Accelerated rotation of a wheel and axle 595 

297 Atwood's machine 599 

298 — 299 Accelerated motion of a system of pulleys or tackle 601 

300 Rolling motion of a body on an inclined plane 605 

CHAPTER II. 

THE CENTRIFUGAL FORCE OF RIGID BODIES. 

301 Normal force 606 

302 Centripetal and centrifugal forces 608 



xxiv TABLE OF CONTENTS. 

§ 303 — 304 Mechanical effect of the centrifugal force 610 

305 — 308 Centrifugal force of masses of finite dimensions. 614 

309 — 311 Free axes, principal axes 634 

312 Action upon the axis of rotation 62& 

313 Centre of percussion ^ ^ 634 



CHAPTEE III, 

of" the ACTION" OF GKAVITT UPOK BODIES DESCRIBING 

PRESCRIBED PATHS. 

314 — 318 Sliding upon an inclined plane 639 

319 Rolling motion upon an inclined plane 646 

330 Circular pendulum 648 

331—333 Simple pendulum 649 

334 Cycloid 655 

335—336 Cycloidal pendulum 656 

337 Compound pendulum 661 

338 Kater's pendulum 664 

339 Rocking pendulum 665 

CHAPTER IV. 

THE THEORY OF IMPACT. 

380—331 Impact in general 667 

333 Central impact .669 

333 Elastic impact 671 

334 Particular cases of impact 673 

335 Loss of energy by impact 674 

336 Hardness of a body 676 

337 Elastic — ^inelastic impact 678 

338 Imperfectly elastic impact 680 

839—340 Oblique impact 683 

341 Friction of impact, friction during impact 685 

343 Impact of revolving bodies 688 

343 Impiipt of oscillating bodies 690 

344 Ballistic pendulum 693 

34^ ^Eccentric. impact 695 

346 Application of the force of impact 696 

347 Pile driving 698 

348 Absolute strength of impact 703 

349 Relative strength of impact 705 

350 Strength of torsion in impact . 707 



TABLE OF CONTENTS. xxv 

SECTION VI. 

STATICS OF FLUIDS 



CHAPTER I. 

OF THE EQUILIBRIUM AND PRESSURE OF WATER m VESSELS. 

PAGE 

§ 351 Fluids 712 

352 Principle of equal pressure 713 

353 Pressure in the water 715 

354 Surface of water 718 

355 Pressure upon the bottom 721 

356 Lateral pressure 724 

357—359 Centre of pressure 725 

360 Pressure in a given direction 731 

361 Pressure upon curved surfaces 734 

362 Horizontal and vertical pressure in water 736 

363 Thickness of pipes and boilers 738 

CHAPTER II. 

EQUILIBRIUM OF WATER WITH OTHER BODIES. 

364 — 366 Buoyant eflfort or upward pressure 742 

367—368 Depth of floatation 746 

369—370 Stability of a floating body 750 

371 Inclined floating 754 

372 Specific gravity 756 

373 Hydrometers, Areometers 758 

374 Equilibrium of liquids of different densities 761 

CHAPTER III. 

OF THE MOLECULAR ACTIOK OF WATER. 

375 Molecular forces 762 

376 Adhesion plates 762 

377 Adhesion to the sides of a vessel 763 

378—379 Tension of the surface of the water 765 

380 Curve of the surface of water 767 

381 Parallel plates 770 

382—383 Capillary tubes 772 

CHAPTER IV. 

OF THE EQUILIBRIUM AND PRESSURE OF THE AIR. 

384 Tension of gases 776 

385 Pressure of the atmosphere. 777 



xxvi TABLE OF CONTENTS. 

PAGE 

§ 386 Manometer 778 

387 Mariotte's law 780 

388 Work done by compressed air 783 

389 Pressure in the dift'erent layers of air. Barometric measure- 

ments of heights 787 

390 Stereometer and volumeter 788 

391 Air pump 790 

392 Gay Lussac's law 793 

393 Heaviness of the air , 795 

394 Air manometer 796 

395 Buoyant effort of the air 797 

SECTION VII. 

DYNAMICS OF FLUIDS. 
CHAPTEE I. 

THE GEKEEAL THEORY OF THE EFFLUX OF WATER FROM 

VESSELS. 

396 Efflux. Discharge 800 

397 Velocity of efflux 801 

398 Velocities of influx and efflux 803 

399 Velocities of efflux, pressure and heaviness 801 

400 Hydraulic head 803 

401 Efflux through rectangular lateral orifices 810 

402 Triangular and trapezoidal lateral orifices 813 

403 Circular orifices 815 

404 Efflux from a vessel in motion 817 

CHAPTEE II. 

OF THE CON^TRACTIOl^ OF THE VEIN OR JET OF WATER, WHEK 
ISSUING FROM AK ORIFICE IN A THIN PLATE. 

405 Coefficient of velocity 820 

406 Coefficient of contraction 821 

407 Contracted vein of water 823 

408 Coefficient of efflux 824 

409 Experiments upon efflux 825 

410 Rectangular lateral orifices, Efflux through them 828 

411 Overfalls 833 

412 Maximum and minimum contraction 834 

413 Scale of contraction 836 

414 Partial or incomplete contraction 837 

415 Imperfect contraction 840 

416—417 Efflux of moving water 842 

418—419 Lesbros' experiments 846 



TABLE OF CONTENTS. xxvii 

CHAPTER III. 

OF THE FLOW OF WATER THROUGH PIPES. 

PAGE 

§ .420 Short tubes 852 

,421 Short cylindrical tubes 853 

. 422 CoeflBcient of resistance 855 

423 Inclined short tubes or ajutages 857 

424 Imperfect contraction 858 

425—426 Conical short tubes or ajutages 861 

427 — 429 Resistance of the friction of water 863 

430 Motion of water in long pipes 869 

431 Motion of water in conical pipes 872 

432 Conduit pipes 874 

433 Jets of water 876 

434 Height of jets of water 878 

435 Piezometer 881 

CHAPTER IV. 

RESISTANCE TO THE MOTIOiT OF WATER WHEN^ THE COKDUIT 
IS SUDDENLY ENLARGED OR CONTRACTED. 

436 Sudden enlargement 883 

437 Contraction 885 

438 Influence of imperfect contraction 88 / 

439 Relations of pressure in cylindrical pipes 888 

440 Relations of pressure in conical pipes 891 

441 Elbows, resistance of 894 

442 Bends 896 

443 — 444 Valve gates, cocks, valves 900 

445 Valves 904 

446 Compound vessels 907 

CHAPTER V. 

OF THE EFFLUX OF WATER UNDER VARIABLE PRESSURE. 

447 Prismatic vessels 910 

448 — 449 Communicating vessels 911 

450 Notch in the side. ^ 914 

451 Wedge-shaped and pyramidal vessels 916 

452 Spherical and obelisk-shaped vessels 919 

453 Irregularly-shaped vessels 921 

454 Simultaneous influx and eflBux 922 

455 Locks and sluices 924 

456 Apparatus for hydraulic experiments 926 



xxviii TABLE OF CONTENTS. 

CHAPTER VI. 

OF THE EFFLUX OF AIR AND OTHER FLUIDS FROM VESSELS 

AND PIPES. „,„, 

PAGB 

§ 457 Efflux of mercury and oil. .;..;.;....,.... 930 

458 Velocity of efflux of air . ... . 932 

459 Discharge 933 

460 Efflux according to Mariotte's law 934 

461 Work done by the heat 936 

462 Efflux of air, when the cooling is taken into consideration 939 

463 Efflux of moving air 941 

464—465 Coefficients of efflux of air 944 

466 Coefflcients of friction of air 949 

467 Motion of air in long pipes 950 

468 iEfflux when the pressure diminishes 952 

CHAPTER VII. 

OF THE MOTIOK OF WATER IN" CANALS AND RIVERS. 

469 Running water 955 

470 Different velocities in a cross-section 956 

471 Mean velocity of running water 957 

472 — 474 Most advantageous profile 959 

475 Uniform motion of water 965 

476 Coefficients of friction 966 

477 — 478 Variable motion of water 969 

479 Floods and freshets 973 

CHAPTER VIII. 

HYDROMETRT, OR THE THEORY OF MEASURING WATER. 

480 Gauging, or the measurement of water in vessels 976 

481—483 Regulators of efflux 977 

484 Prony's method 982 

485 Waterinch 983 

486 Methods of causing a constant efflux 985 

487 Hydrometric goblet «86 

488 Floating bodies 989 

489 Determination of the velocity and of the cross-section 990 

490—491 Woltman's mill or tachometer 992 

492 Pitot'stube 998 

493 Hydrometrical pendulum 999 

494 ^'Rheometer 1001 

CHAPTER IX. 

OF THE IMPULSE AND RESISTANCE OF FLUIDS. 

495—496 Reaction of water , 1002 

497 Impulse and resistance of water 1006 



TABLE OF CONTENTS. ixii 

PAQB 

498 — 500 Impact of an isolated stream 1006 

501 Impact of a bounded stream 1011 

502 Oblique impact 1012 

503 Impact of water in water , 1014 

504 — 505 Experiments with reaction wheels 1015 

506 Water-meters 1020 

507—508 Gas-meters 1023 

509 Action of unlimited fluids 1029 

510 Theory of impact and resistance • • . . 1030 

511 Impulse and resistance against surfaces 1031 

512 Impulse and resistance against bodies 1033 

513 Motion in resisting media 1035 

514 Projectiles 1038 

APPENDIX. 

THE THEORY OF OSCILLATION. 

1— 2 Theory of Oscillation 1043 

3 — 4 Longitudinal vibrations 1045 

5 Transverse vibrations 1048 

6 Vibrations due to -torsion 1050 

7 Density of the earth 1051 

8— 9 Magnetism 1058 

10 Oscillations of a magnetic needle 1055 

11 — 12 Law of magnetic attraction , 1056 

13 Determination of the magnetism of the earth 1059 

14_15 Wave motion 1061 

16 Velocity of propagation of waves 1064 

17 Period of a vibration 1067 

18 Determination of the modulus of elasticity 1069 

19 Transverse vibrations of a string 1070 

20 — 81 Transverse vibrations of a rod 1072 

22 Resistances to vibration 1077 

23 Oscillation of water 1079 

24 Elliptical oscillations 1081 

25—28 Water waves 1084 

Translator's Appendix 1092 

Index 1105 



THEORETICAL MECHANICS 



INTRODUCTION 



TO 



THE CALCULUS. 



Akt. 1 . The dependence of a quantity y upon another quan- 
tity X is expressed by a mathematical formula : e.g., y — dx^, or 
y = a x"\ etc. We write y =f (x) or z = (p (y) etc., and we call y a 
function of x, and z a function of y. The symbols/ and 0, etc., in- 
dicate in general that y is dependent upon x, or z upon y, but leave 
the dependence of these quantities upon one another entirely un- 
determined, and do not give the algebraical operation by which y 
can be deduced from x, or z from y. A function y =f(x) is an 
indeterminate equation ; it gives an unlimited number of values for x 
and y, which correspond to it. If one of them (x) is given, the other 
(y) is determined by the function, and if one of them is changed, the 
other also undergoes a change. Therefore the indeterminate quan~ 
titles X and y are called Variables, or variable quantities ; and the 
quantities which are given, or are to be regarded as given, and in- 
dicate the operation by which y is to be deduced from x, are called 
Constants, or constant quantities. That one of the variables 
which can be chosen at pleasure is called the independent variaUey 
and the other, which is determined by means of a given operation 
from the first, is called the dependent variable. In y—a :c'", a and 
m are constants, x is the independent and y the dependent va- 
riable. 

The dependence of z upon two other quantities, x and y, is ex- 



34 



INTRODUCTION TO THE CALCULUS. 



[Art. 2. 



pressed by the equation z=f {x, y). In this case z is at the same 
time a function of x and y, and we have here two indejpendent 
variables. 

Art. ^, Every dependence of a quantity y upon another quan- 
tity X, expressed by a function or formula y ~ f {x) can be repre- 
sented by means of a curve, A P Q^ Fig. 1 and Fig. 2. 



Fig. 2. 




M N 



The different values of the independent variable x answer to the 
abscissas A M, A N^ etc., and the different values of the dependent 
variable to the ordinates MP, N Q, etc., of the curve. The co-or- 
dinates (abscissas and ordinates) represent then the two variables 
of the function. 

The graphic representation of a function, or the referring of the 
same to a curve, presents several advantages. It furnishes us in 
the first place with a general view of the connexion between the 
two variable quantities ; secondly, it replaces a table or summary of 
every two values of the function belonging together ; and thirdly, it 
affords us a knowledge of the different properties and relations of the 
function. If with the radius C A = C B = r we describe a circle 
AD B (Fig. 3), corresponding to the function y = ^ % r x — x^ 

where x and y indicate the cc-ordinates A M, 
MP, this curve affords us not only a general 
view of the different values that the function 
can assume, but also makes us acquainted 
with other peculiarities of this function, for 
the properties of the circle have also their 
meaning in the function. We know, e.g., 
without farther research, that y becomes equal 
to zero, not only when x = but also when 
a; == 2 r, and that y is a maximum and = r when x = r. 




Akt. 3.] 



INTRODUCTION TO THE CALCULUS. 



35 



Fig. 4 



Art. 3. The Laws of Nature can generally be expressed by 
functions between two or more quantities, and are therefore in 
most cases capable of a graphic representation. 

(1) When a body falls freely in vacuo, we haye for the ve- 
locity y, which corresponds to the height of fall x, y ~ V % g x, 

but this formula corresponds to the equation y — V pxoi the para- 
bola, when the parameter (p) of the latter 
is made equal to the double acceleration 
(2 g) of gravity. We can therefore repre- 
sent graphically the laws of the free fall 
of a body by the parabola A P Q (Fig. 4), 
whose parameter p = 2 g. The abscissas 
AM, AN, of this curve are the space 
traversed by the body in its fall, and the 

ordinates MP, and NQ, the corresponding velocities. 

(2) If « is a certain volume of air under the pressure of one 
atmosphere, we have according to Marriotte's Law, the volume of 

the same mass of air under a pressure of x atmospheres, y = -? 
and we have, for x=l, y = a ; for x ?= 2, y — -, for x = 4:, y = -, 




a 



a 



for a:=10, y=j7:', fora;=100,i/= -— , fora;= co,y=0. 



10 



100' 



We see in this manner that the volume becomes smaller as the ten- 
sion becomes greater, and that if the law of Marriotte were correct 
for all tensions an infinitely great tension would correspond to an 
infinitely small volume. v 

Further, for x=: ^,we have y= 2a ; for x—\, we have y =4:a; 



a 



X = ^^ 



(( 



1 0? 



y—lOa; " x=0. 



u 



y-^a; 



so that the smaller the tension, the greater the volume becomes ; 
and if the tension is infinitely small the volume is infinitely 
great. 

The curve which corresponds to this law is drawn in Fig. 5. 
A M, A N, are the tensions or abscissas x, MP, N Q, the corre- 
sponding volumes or ordinates y. We see that this curve ap- 
proaches gradually the axes A X and A Y without ever reaching 
them. 

(3) The dependence of the expansive force of saturated steam 



36 



INTRODUCTION TO THE CALCULUS. 



[Akt. 3. 



upon its temperature x can be expressed, at least within certain 
limits, by the formula 

and by experiment we have within certain limits a = 75, & = 175, 
and m=6. If we put 

nb-\-xv 

Fm. 6. 



Fig. 5. 





jVi 100 N 20» 



and assume the formula to be correct without limit, we obtain 

-—-I — 1,000 atmosphere, 

^^ = 0,133 
« X = 0», y = (^^J = 0,006 
« X = -75", y = [~^ = 0,000 

^-^j - 1,914 
« ic = 150°, y = (^1^)' = 4,517 
'' x= 200°, y = (^^)' = 15,058 



i( 



a 



(( 



a 



C( 



a 



P Q, Fig. 6, presents to the eye the corresponding curve. It 
passes at a distance A = — 75 from the origin of co-ordinates 



Art. 4.] 



INTRODUCTION TO THE CALCULUS. 



37 



A through the axis of abscissas and at a distance A S= 0,006 
cuts the axis of ordinates ; an abscissa A 31 <^ 100 corresponds to 
an ordinate MP <Cl, and an abscissa A JV y 100 belongs to an 
ordinate NQ> 1 ; ^iid we can also see that not only y augments 
as X increases to infinity, but also that the curve becomes steeper 
and steeper as x becomes greater. 

Art. 4. A function z =f (xy) with two independent varia- 
bles can be represented by means of a curved surface BCD, Fig. 
7, in which the independent variables x and y are given by the 
abscissas A M and ^ A^ on the axes A X and A Y, and the de- 
pendent variable z by the ordinate P of a point F in the surface 
ABC. If for a definite value of x we give different values to y, the 
values of z deduced furnish us with the ordinates of the points of a 
curve ^PP parallel to the co-ordinate plane YZj if on the contra- 
ry for a given value of y we take different values of x, we determine 
the ordinates z of the points of a curve GPU parallel to the co-or- 
dinate plane XZ. We can consequently consider the whole curved 
surface B CD as the union of a series of curves parallel to the co-or- 
dinate planes. The law of Marriotte and Gay-Lussac z = — ^, 



X 



by means of which we can calculate the volume ^ of a mass of air 
from the pressure x and the temperature y, is graphically repre- 
sented by the curved surface C K P H, Eig. 8. A Mi^ the pres- 



Fig. 7. 



Fia. 8. 





sure X, A Nor MO the temperature y, and P the correspond- 
ing volume z : the co-ordinates of the curve P GH give the vol- 
umes for a temperature A N = y, and those of the right line K P 
^e volumes for the same pressure A M = x. 



38 



INTRODUCTION TO THE CALCULUS. 



[Art. 5. 



Aet. 5. When we increase the independent variable of a func- 
tion or the abscissa AM=^x (Fig. 9 and Fig. 10) of the correspond- 
ing curve an infinitely small quantity M JV, which we will in future 
designate by d x, the corresponding dependent variable or ordinate 
MP = y becomes NQ—y', being increased by an infinitely small 
quantity RQ = NQ — MP, to be designated by d y. Both these 
increments dx and dy oix and y are called the Difierentials of the 
Variables or Co-ordinates x and y, and our principal problem now 
is to determine for the functions that most commonly occur the 
differentials, or rather the ratio of the differentials of the varia- 
bles X and y belonging together. If in the function y — f (x), 
where x represents the abscissa A M, and y the ordinate MP, we 
substitute, instead oix, x + dx = AM+ MN— A N, we obtain, 
instead oi y,y + dy = MP + R Q = JSf Q ; therefore 

y ^ dy —f {x + dx), 

and subtracting the first value of y from it, the differential of the 
variable y remains, i. e. 

dy = df{x) = f {x + dx) —f (x) 



Fig. 9. 



Fig. 10. 




This is the general rule for the determination of the differential of a 
function, which when applied to different functions furnishes sev- 
eral rules more or less general : E.G., if y = x^, we have 

d y = (x + dxy — x^ 

{x + d x)^ = x^ -h 2 X d X + d x^ 

d y = 2 X d X + d x^ ~ {2 x + d x) dx; 

and more simply since d x, being infinitely small compared to 2 x, 
disappears, or since 2 x \& not sensibly changed by the addition 
of d X, and the latter can therefore be disregarded, 

dy — d {xf =z 2 X dx* 



N 1 


P 


Q 











D 


C 





Art. 6.] INTRODUCTION TO THE CALCULUS. 39 

The formula y = x"^ corresponds to the contents of a square, 
A B CD, Fig. 11, whose side w> AB — AD — x, 
* ■ and we see from the figure that, by the addition to 

the side of B M—D N = d x, the square is in- 
creased by two rectangles B and D P = %xdx, 
and by a square {d x'), so that by an infinitely 
small increase d x of x the square y = x^ is in- 
creased by the differential quantity 2xdx. 
~BU Art. 6. The right line, T P Q, Fig. 9 and 

Fig. 10, passing through two points P and Q of 
the curve, which are at an infinitely small distance from each other, 
is called the Tangent to this curve, and determines the direction 
of the curve between these two points. The direction of the tan- 
gent is given by the angle P T M — a at which the axes of abscis- 
sas A JTis cut by the line. When the curve is concave, as A P Q, 
Fig. 9, the tangent lies beyond the curve and the axis of abscissas ; 
but when it is convex, as A P Q, Fig. 10, the line lies between the 
curve and the axis of abscissas. 

In the infinitely small right-angled triangle P Q R (Fig. 9 and 
Fig. 10), with the base P R = dx, and the altitude R Q = dy, the 
angle QP R is equal to the tangential angle P TM=a, and we 

have ^ /^ T^ T> Q R 

tang. Q P R =: ^, 

whence . dy 

tang, a = -^^ ; 
^ dx 

therefore the ratio or quotient of the two differentials d y and d x 
gives the trigonometrical tangent of the tangential angle ; e.g., for 
the parabola whose equation isy^=pxwe have, putting y'' = px=Zy 

dz={y + dyy -y' = y' + 2y dy ^ dy' - y' = '2y dy ^ dy\ 

or as dy"^ vanishes before 2ydy, or what is the same thing, dy 
before 2y, 

dz = 2ydy, 
and also 

dz~p{x + dx) —px, 

therefore %ydy=pdx, whence for the tangential angle of the 
parabola we have 

^ dx %y 2xy 2x 



40 



INTRODUCTION TO THE CALCULUS. 



[Art. 7. 



The definite portion P T of the tangent between the point of 
tangency P and the point T where it cuts the axes of abscissas 



Fia. 12. 



Fig. 13. 




M N 



X A 




M N 



is generally called the Tangent, and the- projection T M oi the 
same upon the axes of abscissas the 8uh-tangent ; hence we have, 

suUang. ^ PMcot. P T^3I 

d X 



=- y cot. a ~y- 



2x 



dij 



E.G., for the parabola, subiang. — y — = %x. 

The subtangent is therefore equal to the double abscissa, and 
from it the position of the tangent for any point P of the para- 
bola is easily found. 

For the curved surface BCD, Fig. 7, the angles of inclina- 
tion a and /3 of the tangents P T and P C7 at a point P are 
determined by the formulas : 

d z J „ d z 

tana, a = -z— tana, p = ^r— 

^ dx '^ dy 

The plane P T U passing through P T and P U is the tan- 
gent-plane of the curved surface. 

Art. 7 , For a function y — a + mf {x) we have 

dy =\a + mf {x ■\- dx)] — [« + mf (xy] ; 
= a — a -[- mf (x -\- dx) — mf (x 
= m[f{x + dx) —f{x)']; 
i. e. 

I.) d\a + mf {x)'\ = mdf {x), 

E.G., ^ (5 + 3^') = 3 [{x + dxY —x'] = S.2xdx = Qxdx, 
In like manner : 
d{4:- ix'), ^ -id{xy = -i [{x 4- dxf - x'] 

= — i (x'^ + 3x'^ dx + dxdx^ + dx" — x^) 
— ^ ■"— 7y • o »</ ax — '7) X a X. 



Aet. 7.] 



INTRODUCTION TO THE CALCULUS. 



41 



Hence we can establish the following important rule : The con- 
stant member {a, 5) of a function disappears by differentiation, and 
the constant factors remain unchanged. 

The correctness of this rule can be graphically represented. 
For the curve A P Q, Fig. 14, whose co-ordinates in one case are 



Fig. 14 



Fig. 15. 




Ml Ni 




AM — X and MP = y — f {x)^ and in the other A^M^ —x and 
Ml P = a ^ y = a + fix), we have P R — d x imd R Q = d y = 
df ix) and also = d {a -{- y) = d [a -h f (x)]; and for the curves 
A Pi Qi and A P Q, Fig. 15, whose corresponding ordinatcs M P^ 
and M P as well as iV^ ft and N Q have a certain relation to one 
another, the relation between the differentials Ri Qi — N Qi — 
MPi and R Q = JVQ — MP is the same ; for if we put MP^ = m . 
MP 8ind]^Qi = m. JVQ,it follows that Ri Qi = NQi ~ MPi^m. 
{NQ -MP) =m . Q R. 

i. e. d [mf (x)] =mdf {x). 

If y = ti + V, or the sum of two variables u and v^ we have 

d y = tc + d tt + V -{■ d V — {to -\- v), i. e., according to Art. 5. 

II.) . . . d (21 -{■ v) = du + dv, and in like manner, 
d [f(x) + (x)] = df{x) + dcf) (x). 

The differential of the sum of several functions is then equal 
to the sum of the differentials of the separate function ; e.g. 

d{2x + Sx"" — ^x^) = 2dx + ^xdx — ^x"" dx = {2 + 6x — -^x'')dx. 

The correctness of this formula can also be made evident by the 
consideration of the curve A P Q, Fig. 15. If M P=f (x) and 
P Pi = (f) (^x) we have 

MPi = y —f {x) + {x) and 

dy = RiQi = RiS+SQi-:^RQ-¥SQi = df(x) + d(h{x)'. 



42 INTRODUCTION TO THE CALCULUS. [Art. 8. 

for Pi S can be drawn parallel to P Q, and therefore we can put 
RiS^BQand Q S = P P,. 

Aet. 8. If y = u V or the product of two variables, e.g. the 
contents of the rectangle A B CD, Fig. 16, with the variable sides 
AB — u and B C = v, we have 
dy = (u -\- du){v + dv) — uv — uv ^ itdv + vdu + dudv—uv, 

— udv + vdu + dudv = udv + {v + dv) du. 

But m V -^ dv, dv \^ infinitely small com- 

FiG. 16. -I , J ^ , 

pared to v, and we can put 

^^^^^■111 V + dv = V, and {v + dv) d u = v d u, 

D ^IJI and also 

jll udv + (?; + dv)du — udv 4- ^^(^«^, 

~~ bUi so that 

III.) . . . d{uv)~ udv -\- vdu, 
it follows therefore that 

d\f{x).<\> {x)'\ =f{x) d^(x) + (f) {x) df {x). 
The differential of the product of two variables is then equal to 
the sum of the products of each variable by the differential of the 
other. 

When the sides of the rectangle A B C D are increased by 
BM—du and D — dv\i^ contents y = AB x AD = uvis aug- 
mented by the rectangles C = udv and C M = v du and C P 
= dudv, the latter, being infinitely small, compared with the oth- 
ers, disappears ; the differential of this surface is only equal to the 
sum udv + vduofthe contents of the two rectangles C and CM. 
In conformity with this rule we have for y — x (Sx'' + 1) : 
dy=--xd{dx' + 1) -^{Sx' -^ l)dx = 3xd(x') +(3:^;^ 4- 1) dx 
= Sx.2xdx-\-Sx''dx-hdx={9x''-hl) dx. 
Further, if w be a third variable factor, we have 

d (uv w) — ud{v w) 4- V IV du, 
or since d {v w) = v d w + w d v, 

d {u V w) = u V d w + u w d V -\- V tv d u, and in like manner 
d{uvwz) = uvwdz + u v z d w -^-u w z d v -{- v w z d u ; 
if w=v=w=z, it follows that d (u*) =4:U^ du, and in general 
lY.) , . . d {x"')=m x'"'' dx, if m is a positive integer, e.g, 
d{x'') = 7x'dx, d^x'^Qx' dx. 
Ify— x~''\ m being again a positive integer, we have also 
yx'^ = l and d (y x"") = 0, i. e. 
y d iptf") -^-x"^ dy — 0, and therefore 

, udix''') x~'"mx'^~^ dx ,^, , 

dy^—^ — ^^ — <- — = — mx~'^^ dx. 

V rpoi rf.li. ' 



Art. 8.] INTRODUCTION TO THE CALCULUS. 45 

or, if we put — m = n, 

The Eule IV. applies also to powers, whose exponents are neg- 
ative whole numbers, as e.g., 

d(x~^) = — 3 x~* dx= r-j and 

^ ' X 

d(3x^ + 1)- =.-2{3x^ + l)-^d{Sx^) = - ^3^^. 

If in ?/ = a;", ^ is a fraction whose denominator n and whose 

numerator 7n are integers, we have also y" =x'^ and d (^") = d (a;"*), I.E., 
n y""'' dy = mic'"~^ dx, therefore 

, _ mx'^~^dx_m x^~^ dx _m --\-j 

d if — ^ " ;; , — ~; — X n Ct Xt 

^ n y n ^^'1 n 

X n 

If we put — — », it follows that 
n 

dy = d{x^) — p xP~^ dx, which agrees with Eule IV., which 
can now be considered as general. 

Also d ( u^) = p u^~^ d u, when w denotes any function de- 
pendent upon x. 

Hence we have, E.G., d{V x^)—d {xi)=^x'i dx=^ Vx^d x, 

d V'Zrx—x'—d\^=d{u^)—^u~^du 

_ I d {2 rx — x"^) _ 2 r dx — 2xdx _ (r — x) dx 

u^ 2 \/^ i/2 rx-x" 

u 
In order to find the differential of a quotient y = -,we put u = 

V y, whence du = vdy-\-ydv, and 

ii 

', I.E., 



, , dti av 

dy= ^u—ydv^ ^ v 



^•) <h 



V V 

vdu—udv 



v' 



According to this Eule, E.G., 
^/^'-1\ _ {x + 2)d (x' - 1) - (x'- l)d(x-^ 2) 



{x-\-2y 

. dx _ 



^(x + 2) .2xdx - {x' - l).dx _ / x' + ^x + l \^^ 

\ (x -{- 2y I ' 



44 INTRODUCTION TO THE CALCULUS. [Abt. 9, 

We have also : . ' 

\ V / v" ' Xx" I x" X" 

Akt. 9. The function ^ = ^" is the most important in the 
whole analysis, for we meet it in all researches. When we give the 
exponent n all possible values, positive and negative, whole and 
fractional, etc., it furnishes the different kinds of curves, which are 
represented in Fig. 17. A is here the point of origin of the co-ordi- 
nates, XX the axis of abscissas, and Y J" that of the ordinates. 

If on both sides of the co-ordinate axes at the distances x = ± 1 

and y = ±1 from the point A we draw the parallels Xi Xi, Xs Xg, 
Yi Yi, Y2 Y2 to the axes, and join the points Pi, Pg? Ps^ and Pi^ 
where they cut each other, by means of the diagonals ZZ, Z^ Z^^ we 
obtain a diagram which contains all the curves, given by the equa- 
tion y ~ x'\ For every point on the axis of abscissas X X we have 
«/ = 0, and for every point on the axis of ordinates Y Y, x = 0; 
and for the points in the axes Xi Xi and Xg X2 , «/ = ±1, and 
for the points in the axes Yi Yi and Y^ Y^, x= ±: 1. 

If in the equation y = x'' we put x ~ 1, we obtain for all 
possible values of n, y = 1, and for certain values of n, also 
y = — 1; consequently all the curves belonging to the equa- 
tion y — x"" pass through the point Pi, whose co-ordinates are 
A M = 1 and ^ X = 1. If we take n = 1 we have y = x and we 

obtain the right line Z A Z, which is equally inclined to the two 
axes XX and Y Y, and which rises on one side of A at an angle 

of 45° ( J 1, and on the other side dips at the same angle. On the 

contrary, for 2/ = — xwe obtain the right line Z^ A Z^ which dips 
on one side of A at an angle of 45°, and rises on the other side at 
the same angle. 

If, however, n"> 1, y — x'' becomes smaller for x <Z 1, and for 
ic > 1 greater, than x, and when 71 <^l,y = x" i^ greater for a; < 1 
and smaller for x y> 1 than x. The first case {n y 1) corresponds 
to convex curves, which run in the beginning under, and from Pi 

over the right line {Z A Z), and the second case (n < 1) to concave 
curves, where the reverse takes place. 

When, in the first case, we take n smaller and smaller until at 
^st it disappears, or becomes equal to zero, the ordinates approach 



Art. 9.] INTRODUCTION TO THE CALCULUS. 45 

the constant value y = x^ — 1 and tlie corresponding curve ap- 
proaches more and more to the broken line A N P^ Xi'^ if, on the 
contrary, in the second case, n becomes greater and greater, the 
values of the ordinates approach the limit y = x^ = xh = oo, and 

Fig. 17. 




those of the abscissas, on the contrary, approach the value x=y^=l, 
and the corresponding curve approximates more and more to the 
broken line A M F^ Y^. 

If we take n= — 1, whence y = x~^ = -, for x = 0, we have y 

— 00 , and for :r = cc , y = and we obtain curve, which has been 

discussed in Art. 3, and drawn in Fig. 5 ( 1 P 1); it approaches 
on one side the axes of ordinates, and on the other the axes of ab- 
scissas without ever reaching them. 

If the exponent (—?^) of the function y = x~'' =— is a proper 



46 INTRODUCTION TO THE CALCULUS. [Aet. 9. 

fraction, for ^ < 1, we have y <^~, and on the contrary for a; > 1, 

2/ >-, and if this exponent is greater than unity, we have on the con- 

trary for a; < 1, ?/ > -, and for a; > 1, ?/ < -• The curve corre- 
al X 

sponding to y = x~'\ according as n is greater or smaller than unity, 
runs in the beginning below or above, and from P^ above or below, 

the curve y — x~^ = -. While those curves, which correspond 

X 

to the positive values of n, are placed in the beginning below, 
and from Pi on above, the right line Xi Xi, the curves of the nega- 
tive exponents (— n) run first above, and from P^ on below, Xx Xi. 
For the former curves we have, for ?/ — 0, i?; = 0, and for x = cc, 
y = CO , and for the latter, for x = 0, y = oo , and for ic = cx), 
y = 0. While the former diverge more and more from the co-or- 
dinate axes X Jr and Y Y, the farther we follow them from the 
origin A^, the latter approach more and more on one side the axis 

X Xf and on the other axis Y Y, without ever reaching them. 

The last system of curves approach nearer and nearer the 
broken line Y X Pi Xi or the broken line Yi Pi M X'dB the expo- 
nent approaches nearer and nearer the limit n = or n = co. 

If in y = x"^"" , m is an entire uneven number (1, 3, 5, 7 . . .), ?/ 
and X have the same sign. Positive values of x correspond to positive 
values of y, and negative values of x to negative values of y. If on 
the contrary m is an entire even number (2, 4, 6, etc.), y becomes 
positive for all values of x, positive or negative. Therefore the 

curves in the first case, as e.g., (3 Pi ^ Pg 3) or (1 Pi 1, T Pg 1), 
run on one side of the axis of ordinates above, and on the other side 

below, the axis of abscissas X A Xj on the contrary the curves in 

the second case, as e.g., (2 Pi J. P4 2) or (2 Pi 2, 2 P4 2), are placed 
above the axis of abscissas only, and are contained in the first and 
fourth quadrants ; the former corresponds for m = ± 00 to the limit- 
ing lines YiMA Ml Y^ and XM Yi, X Mi 1^, the latter on the 
contrary to the limiting lines Yi M A Mi Y^ and X M Yi 

XM Y,. 

If we have y — x^'% n being an entire uneven number, y and 
x have the same signs, and if n is an entire even number, every 
positive value of x gives two equal values for «/, one of which 



Art. 10.] INTRODUCTION TO THE CALCULUS. 4? 

is positive and the other negative, and on the contrary for every 
negative value of x, y is imaginary or impossible. The curves, 
as E.G. (I Pi A Pz I ), which correspond to the first case, are found 
only in the first and third quadrants, and the curves of the second 
case, as e.g. (^ Pi A P^ h), only in the first and second quad- 
rants: the former become for m = oo the limiting lines JTj N 

A Ni X2 and Xi N Y, X^ Ni Y, and the latter the limiting lines 

Xi NA Ni X, and Xi N Y, X, Ni Y 

Since y = x^" involves x = ?/^% it follows, that the latter sys- 

tern of curves \^ = x '7 differs from the former (y = x'^'^) in its 
position only, and that by causing them to revolve, the curves of 
one system may be made to coincide with those of the other. 

m / 1 \ot 1 

Since y = x" ~ \x"J = {x'^'Y we can always give from what 
has gone before the general course of a curve, e.g., the curve 
for 

has, for both positive and negative values of x, positive ordinates j 
on the contrary, the curve for 

y = x^ = (xiy = (f^xj 

has, for positive values of x only, real ordinates, and they are equal 
in magnitude, but with opposite signs. Further, for the curve 



y = xi = \yxy 



y and x have the same sign, since neither the fifth root nor the 
cube causes a change of sign. 

Finally, the curves, which correspond to the equation y — 

—X", differ from those of the equation y — x"" only by their reversed 

position in regard to the axis of abscissas X X, and they form the 
symmetrical halves of a complete curve. 

Art. 10. From the important formula d (af) = n x""-^ d x we 
obtain the formula for the tangential angle of the corresponding 
curves represented in Fig. 18. It is 



48 



INTRODUCTION TO THE CALCULUS. 



[Art. 10 



and therefore we have the subtangent of these curves 

d X oif X 



dy n af ~^ n 
Hence, for the so-called parabola of Neil, the equation of which 

— y —, we have 



is Cb if =■ ^ or y 



1 d{7^) __!_ , ^ J X 



a 



and the subtangent — | ic. 



a' 



Farther, for the curve already discussed y = ~ = a^ x~\ 



X 



tang, a = a" —^ — '- = ^ = — ( - I , 

d X x^ \ X / 



X 



and the subtansfent = -^ = — x. (See Fig, 5.) 



-1 



Fig. 18. 




Art. 11.] 



INTRODUCTION TO THE CALCULUS. 



49 



Consequently, we have for x = 0, tang, a = — co and a = 90°, 

for X = a, tang, a = — 1 and a =135° 
and for x= cc, tang, a = and a =i 0°, etc. 
Art. 1 1, When a right line A 0, Fig. 19, cuts the axis of ab- 
scissas at an angle A JC = a, and is at a distance C K — n from 
the origin of co-ordinates C, the equation between the co-ordinates 
C M — N P = x and CN=MP — yoi^ point in the same is 
y COS. a — X sin. a = n, since n = MR — ML, MB = y cos. a 
and M L =^ X sin, a. 



Por X = 0, y becomes C B 
h COS. a, and y cos. a 



b = 



n 



-, therefore we have n= 



COS. a 
X sin. a ^= h. cos. a or 
y — h + X tang. a. 
Generally the lines C A and C B, which measure the distances 

from the points where the line cuts 
the co-ordinate axes C X and C Y 
to the origin of co-ordinates, are 
called the jparameters of the line, 
and are designated by the letters a 
and 1). According to the figure 
C A ^ — a, therefore 

h 




CB 

tang, a = ^-j = 



a 



the straight line becomes 



*-, h X 

y = x.ov 

^ a ' a 



and consequently the equation of 



2/_ 



-t- ^ = 1. (See Ingenieur, page 164. ) 

When a curve approaches more and more a line, which is sit- 
uated at a finite distance from the origin of co-ordinates, without 
ever attaining it, the line is called the Asymptote of the 
Curve. 

The asymptote can be considered as the tangent to a point 
of the curve situated at an infinite distance. Its angle of inclina- 
tion to the axis of abscissas can be determined bv 

dy 



tang, a = 



dx' 



and its distance n from the origin of co-ordinates by the equation 
n = y COS. a — X sin. a = (y — x tang, a) cos. a 
y — X tang, a 



Vl 4- {tang, a) 






50 



INTRODUCTION TO THE CALCULUS. 



[Art. 13. 



as well as by the formula n=(ii cot a. a—x) sin. a =- ^ ^' 

^ ^ ^ n + icotg.ay 



-(yry-')W. 1 + © 



when we substitute x and y = co in them. 

In order that a tangent to a point infinitely distant can be an 
asymptote, it is necessary, that for x oy y ~ cf^, y — x tang, a or 
y COS. a — x shall not become infinitely great. 

For a curve whose equation i^ y — x~'"' = — 

X 

m ^ , m m + 1 
tang, a = — --^^^^ and y — x tang, a = x'"" H ~ = 



X 



X' 



X" 



T X 

and also y cotq- a ~ x = — x = — (m -r 1) — , therefore 

1) for ic = 00 , ^ =r 0, tang, a =z 0, y — x, tang, a = and n = 0, 
and 2) for y = oo, x = 0, tang, a — oo, y cotg. — cc = and n = 0. 

The axis of abscissas X X corresponds to the conditions tang, a 

= 00 and n = 0, the axis of ordinates Z" I^ to the conditions 
tang, a = and n = 0; therefore these axes are the asymptotes 
of the curve, corresponding to the equation y = x~'^. (Compare 

the curves 1 Pi 1, 2 F,¥, and ^ Pi \ in Fig. 18, page 48.) 

Art. 12. The equation of an ellipse A D A^ Pi, Fig. 20, can 

be deduced from the equation 

x"" + y''i*= a"" 

of the circle A B A^ B^, whose ra- 
dius i^CA=^CB=^CP^a 
and whose co-ordinates are CM 

— x and M P = yi, when we 
consider, that the ordinate M Q 

— y of the ellipse is to the ordi- 
nate M P = yioi. the circle, as the 
lesser semi-axis G D = h of the el- 
lipse is to the greater semi-axis, 
which is equal to the radius of the 
circle C B = a. We have then 




y^ 



= -, whence y^ = - y and ic' + — ^' = a% i.e. 



Art. 13.] 



INTRODUCTION TO THE CALCULUS. 



51 



-^ -f -^ = 1, the equation of the ellipse. 

(T 



If we substitute in this equation for + 5^ — If, we obtain the 
equation —3 



a' 



1^-1 
h" ~ ^' 



which is that of the hyperbola formed by the two branches P A Q 
and F, A, Q,, Fig. 21. 
When in the formula 

I 

y 



V¥^ 



a 



a 



deduced from the latter equation we take x infinitely great, a^ dis- 
appears before x", and we have 

y :=: ~ y X — ± ■ — — ± X tang, a, 

the equation of two right lines C U and C F passing through the 
origin of co-ordinates C. Since the ordinates 

a a a 




tend to become equal as x becomes greater, it follows that the right 
lines C U and C V are the asymptotes of the Hyjjerbola. 

If we take C A = a, the perpendicular A B = + b and 
A D = — Z», we can determine the two asymptotes; for the tan- 
gent of the angle ± a, formed by the asymptotes with the axis of 
abscissas, is 

tang. A C B = 77—7, i.e. tang, a = -, and 
in like manner 

tang. A C D = 77—7? i.e. tang. (^ a) = . 

If we take the asymptotes U U and F F as axes of co-ordi- 



52 



INTRODUCTION TO THE CALCULUS. 



[Art. 13. 



nates, and put the abscissa or co-ordinate C N in the direction of 
the one axis = u, and the ordinate or co-ordinate iV^ P in the di- 
rection of the other = v, we have, since the direction of u varies 
from the axis of abscissas by the angle a, and that of v by the 
angle — a 

C M =^ X = C JV COS. a + JV P cos. a = (u -h v) cos. a, and 
MP = y = C I^ sin. a — ]Sf P sin. a — (u — v) sin. a. 

If we designate the hypothenuse C B — V a^ + If \ij e, 

we haye 

., cos. a 
and consequently 



a ^ . h 

COS. a ■=. — and s%n. a = — , 
e e 



a 



sm. a 1 , 
— = — = — , and 
e 



-^ — 4^ = -^ COS. a — sm.^ a 

a 



If a' 

u" -\- 2 U V + v"" 



u^ — 2 u V + v"^ 



4:UV 



= 1. 



e e e 

From the latter we obtain what is known as the equation of the 
hyperbola referred to its asymptotes 

u V = — - or ^; = 1 — . 

According to this it is easy to draw the hyperbola between the 
two given asymptotes. 

The co-ordinates of the vertex A are G E — E A =^ —, and 

Fig. 33. 




-u 



the co-ordinates for the point K 2irQ C B = e and B K — -^\ fur- 

B S 

ther, for the abscissas 2 e, 3 e, 4 e, etc., the ordinates are 5 -j-W X' 



e 
1 



I — , etc. 



Art. 13.] 



INTRODUCTION TO THE CALCULUS. 



53 



d v 
Aet. 13. If in the ratio of the differentials -~, or in the for- 

dx 

mula for the tangent tang, a of the tangential angle, we substitute 
successively the different values of x, we obtain all the different po- 
sitions of the tangent to the corresponding curve. If we take x=0, 
we obtain the tangent of the tangential angle at the origin of co- 
ordinates, and if on the contrary we take i?; = oo, we have the same 
for a point infinitely distant. The most important points are those 
where the tangent to the curve runs parallel to one or other of the 
co-ordinate axes, because here one or other of the co-ordinates x and 
y have their greatest or smallest value, or, as we say, is a maximum 
or minimum. When the curve is parallel to the axis of abscissas we 
have a = 0, and tang, a = ; when parallel to the axis of ordinates 
a — 90°, or tang, a = oo , whence we deduce the following Eule : 
To find the values of the abscissa or independent variable x, 
which correspond to the maximum or mini- 
mum value of the ordinate or dependent va- 
riable y, we must put the ratio of the differ- 
dy 



Fig. 23. 




entials 



d X 



— 0, or — 00 and resolve the result- 



ing equation in regard to x; e.g., for the 
equation y = 6 x — -^ x"^ -h x^, which corre- 
sponds to the curve A P Q E in Fig. 23. 



dy _ 



d X 



= e — 9x-\-3x''=^d{2-dx-\-x') = 



d{l ~ x) (2 - x); 

consequently, in placing —^ = 0, we have 

1 — a; = and 2 — a; = 0, 
I.E. X = 1 and X = 2. 

Substituting these values in the formula 
y = 6 X — -^ x"" -j- x% 
we have the maximum value ot y, M P = 6 — 
the minimum value, iV^ § - 12 — 18 + 8 = 2. 

Farther, for the curve K P Q P, Fig. 24, whose equation is 

y = X + ^ {x — 1)^ we have 
dy . . . , .. , . 2 



+ 1 = 



5 



and 



d X 
which becomes 



tang, a = 1 + | (re — 1)-^ = 1 + 

2 



0, for 



= — 1, I.E. ioY AM — X = \ 



Z^x-l 

{ly = if = 0.7037, and on the contrary =: oo, for ^ J\r = a; = 1. 



54 



INTRODUCTION TO THE CALCULUS. 



[Abt. 14. 



The first case corresponds to the maximum value, 

MP:=y^=.l- (f)^ + {^ ^ 31 ^ 1.148, 
and the last to the minimum value, N Q — y^ = 1, 

We have also A =y 
:= 1 for a; = 0, and y = 
for the abscissa A K = x, 
corresponding to the cubic 
equation x^ + x"^ — 2 x -i- 1, 
whose value isx= — 2.148. 
Aet. 14. Since in the 
equation of a curve which 
starts from the origin of 
co-ordinates A, and rises 
X above the axis of abscissas, 
y increases with x, d y is 
always positive, and since 
when the curve on the contrary descends towards that axis, y de- 
creases when X increases, d y becomes negative. Finally at the 
point where the curve runs parallel to the co-ordinate axis A X, 
d y becomes equal to zero, and the differentials of the ordinates, 
corresponding to the equal differentials d x — M N — N = P 8 
= Q T of the abscissas, are 

S Q = P S tang. Q P 8, i.e., dyx — dx tang, a^, 
T R = Q T tang. R Q T, i.e., d y<i = d x tang, a^ , etc. 
The tangential angles a^, a^, etc., also increase for a convex 
curve APR, Fig. 25, and decrease for a concave curve APR, 





Fig. 24. 












/f 




T 


A 


/ 




^^^^ 


] 


> 


/ 




/ 




s 




K/l 













_2 



_1 



A 



M N 



+ 2 



Fig. 25. 



Fig. 26. 



Fig. 27. 









/ 


"a 


/ 


T 


R 


/ 


s 




■» 


T > 


sT ( 


) 




N O 




N O 



Fig. 26 ; consequently in the first case 

d {tang, a) — d Vn--) is positive, 

and in the second d {tang, a) — d(j^\ is negative, and for the 
points of inflexion Q, Fig. 27, i.e. for the places Q where the con- 



Art. 14.] 



INTRODUCTION TO THE CALCULUS. 



55 



vexity changes into concavity, or where the contrary takes place, 
-we have S Q = T R, and therefore d {tang, a) — d \^-) = 0. 

Hence we have the following Eule : 

If the differential of the tangential angle is positive, the curve is 
convex, if it is negative, the curve is concave, and if it is equal to 
zero ive have a point of inflexion of the curve to deal ivith. From 
the foregoing we can easily make the following deductions : 

The place, where the curve runs parallel with the axis of abscis- 
sas and for which tang, a = 0, corresponds either to a minimum or 
to a maximum, or to a point of inflexion of the curve, according as 
the curve is convex, concave, or neither, i.e., as d {tang, a) is pos- 
itive, negative, or equal to zero. On the contrary, the point, where 
the curve runs parallel with the axis of ordinates and for which we 
have tang, a = oo, corresponds to a minimum, or maximum, or to a 
point of inflexion of the curve, according as the latter is concave, 
convex, or in part concave, or in part convex : i.e., as d {tang, a) 
is negative or positive on each side of this point, or has a different 
sign on different sides of it. 

A portion of a curve with a point of inflexion of the first kind 
is shown in Fig. 28, and a curve with one of the second kind in 
Fig. 29. We perceive that the corresponding ordinate JV Q is nei- 
ther a maximum nor a minimum, for in this case both of the 
neighboring ordinates M P and R are larger or smaller than 
N Q. In Geometry, Physics, Mechanics, etc., the determination 



Fig. 28. 




M N O 




Fig. 30. 




M N O 



of the maximum and minimum, or the so-called eminent, values of a 
function, is often of the greatest importance. Since in the course 
of this work various determinations of such values of functions will 
be met with, we will here treat only the following geometrical 
problem. 

To determine the dimensions of a circular cylinder A N, Fig. 
30, which for a given contents V has the smallest surface 0, let us 



56 INTRODUCTION TO THE CALCULUS. [Aet. 14 

designate the diameter of the base of the cylinder by x and the 
height of the same by y ; here we have 

V=-~x'y 

and the surface or the area of the two bases plus that of the 
curved portion 

= — ^— + iTxy, 

but from the first equation we have 

4 V 

TT ^ = — - or n X y = 4: Vx"^ 

substituting this value of tt x y, we obtain 

= ^ + 4 rx-\ 

and since we can treat and x as the co-ordinates of a curve, we have 

tana, a = - — = n x — 4: V x~^. 
^ d X 

Putting this quotient equal to zero, we obtain the equation of con- 
dition 

4 F 

7T X = —^ or 7T X^ = 4: V. 
X 

Kesolving the equation in reference to x, we have 



J/-^^ 



X — y , and 

4 V l/~^rr' '^ l/Tv 

^ 7T X^ ^ 7T^ 16 V ^ 7T 

Since d {tang, a) = I n + — ^J dxis positive, the value found 

furnishes the required minimum. We can employ the same 
method when we wish to determine the dimensions of a cylindri- 
cal vessel which for a given contents will need the smallest amount 
of material. They are already determined directly when the vessel 
besides its circular bottom is to have a circular cover, but when 
the latter is not needed we have 



TT X"^ 



= —T — h 4 F a;~', consequently 

IT X 4 F" 

— - = — 5-, whence it follows that 

2 X 

V~F VF' tt' ^/~V 

^ = ^ y — and «/ = y --r . -^ = y —- = ^ a;. 



n 



n' V' ^ 7T 



Art. 15.] INTRODUCTION TO THE CALCULUS. 57 

While in the first case we must make the height equal to the 
width of the cylinder, in the second we must make it but one-half 
the width of the latter. 

Aet. 15, By successive differentiations of a function y =f{x)f 
we obtain a whole series of new functions of the independent va- 
riable X, which are 

- , , dy df{x) 
'' ^ ' a X ax 

/; (x) = •^-, , fz (x) = -^-r ^ , etc., 
•^^^ dx -'^ ' dx ' ' 

E.G., for y = f (^) = ^K ^^ liave 

/, {x) = i xl,f, (x) - -V- x-h,f, (x) = -i^ x-% etc. 

For a function which is developed according to a series of the 
ascending powers of x 

y = f (x) = Ao + Ai X + A<i x"" + Az x^ + A^ x" + etc., we have 

/i (a;) = ^1 + 2 ^2 ^ + 3 A^x"" + 4 ^4 ic^ + etc. 
fc^{x) = 2A^ + 2.SAzX + 3.4:AiX'' + etc. 
f/{x) = 2 .3 As + 2 .3 . 4: A^x + etc. 

Substituting in these series a; = we obtain a series of expres- 
sions suitable for the determination of the constants Ao, A^, A^.,. viz. 

/ (0) = A,, f, (0) = 1 A„ f, (0) = 2 A„f, (0) = 2 . 3 . ^3, 
etc., whence we deduce these co-efficients themselves. 

A,=f{0), A, =f, (0), A, - if, (0), A, :=: ^3/3 (0), 

^4 = 234 /4 (0) etc. 

Thus we can develop a function into the following series, known 
as McLaurin's. 

fix) = /(O) + /. (0) . f + fi (0) • ^ + /3 (0) ''' 



1 •"' ' 1.2 ■"^' 1.2.3 



+ /^(°)- 172^4 + 



For the binomial function y = f (x) = (1 + ic)" we have 
/, (x) =n{l + xY-\f, {x) =:n{n -1){1 + xy-' 
/g (x) = n {n — 1) (n — 2) {1 + xY'^ etc. 

When we put a; = 0, we obtain 
/(O) = 1,/, (0) = n,f, (0) =^n(n--l) 
fz (0) = n {n — 1) (n — 2), etc., whence the binomial series. 



58 INTRODUCTION TO THE CALCULUS. [Art. 15. 

T^ /•. x„ -. ^ n(n—l) , n(n — l)hi — 2) , 

I.) (1 4- ic)" = 1 + - a; + —Y2 "^ j^ ^ 3 X + etc. 

We have also 

/-. x« -. ^ n(n — l)^ n(n — l){n — 2). 

(1 - a;)'^ = 1 - -^ a; + ^^ ^ ^ o;^ -J-^^ ^ "^ ^^^'^ 

as well as ' 

Farther, puttins' 1 + a; — (1 — 2;)"^ — , we have;2 = - and 

1 — z 1 -t-x 

/-. x« /-. X « -I , ^ (^^ + 1) 2 n(n + l)(7i + 2) 3 

I . /V i. , /i . O 

11.) (1 + ic)" = 1 + Y r^ — + , ^ — ^ h^ — 

^ ^ ^ 1 \1 + 2^7 1.2 M + iz;/ 

^ (^ + 1) (^ + 2 ) / X V 

1.2.3 \r+d ^ * • 



+ 



The series I. is finite for entire positive values of n, and the 
series II. for entire negative values of the same. 

E.G., (1 + a;)' = 1 + b X + l^x" ■\- 10 x^ -^^ b x^ + x\ and 
Since a + ^ = « jl + -1, it follows also that 

(« + .)"=«-(! + 3"= ."[i + f(^) - 

III.) (a + oj)" = a" 4- J a"-^ rr + ^^LZi] ^n- i^« 

^ (^^ - 1) (^ - 2) 



Art. 15.] INTRODUCTION TO THE CALCULUS. §9 

.KG., ^lOOO"^ = (1000 + 9)i = 100 (1 + 0,009)1 

= 100 (l + I . 0,009 + ^^2~^- • (0,009)" + . . . ) 

= 100 (1 + 0,006 - 0,000009) = 100,5991. 
We have also 

{x -\- ly = x"" + n X"-' + \ ^ x""-"" + . . . etc. 

and approximately for very great values of x, 

{x -{- ly — x"" -\- n x''-\ 
From tills it follows that 



^ 


n 


{X- 




{X- 


„.„_., (« - 1)" - (« - 3)" 


-) - « 


(X- 


„^._, {x-%r-{x-3r 


"> - n 


• 


:=: 




1..-. _ 3" - 1" . 



and finally 

adding the two members of these equations together, we have 
of'-' + {x - 1)"-' + {x — 2)"-' -\- {x - 3)"-' + . . . + 1 

_ {x + 1)" — 1" 
~ n ' 

or, putting n — 1 = m, and writing the series in the reversed 
order, we have 

(x 4- 1V"+^ — 1 

im _^ 2"^ -h S"" + . . . + (X — l)"^ + XT = ^ , . 

^ m + 1 

Now since x is very great, or properly infinitely great, we can 
put (x + 1)""+^ = x"'+\ and we then obtain the sum of the powers 
of the natural series of numbers. 



/^W + l 



IV.) I'" + 2"^ + 3"* + . . . + iz;"* ^ — -, 

m -f 1 

E.Q., v^l" + Vy* 4- l/y^ ^ Vl^ + . . . + ^1000' approximately 

= 1^ ^ I Viooo^ := 60000. 

3 



60 



INTRODUCTION TO THE CALCULUS. 



[Akt. 16. 



Fig. 31. 







: 


? 


E^ 


^ 






D^ 










a 


^ 




K 








B, 


/^ 




H 










^ 




G 












A 1 


? ] 


L 1 


kl ] 


^ 




( 


3 



Art. 1 6. The ordinate P = y, Fig. 31, corresponding to 
the abscissa A = x, can be considered as composed of an infinite 

number of unequal elements d y, as 
FB, a G, HD, KE , which cor- 
respond to the equal differentials dx=^ 

AF, = FL^LM= MN of 

the abscissa. If therefore d y — (^{x) . 
d X were given, we could determine y 
by summing all the values of d y, which 
we obtain, by substituting successively 
in {x) d X ioT X, d X, "H d X, ^ d X . . . , 
to n d X = X. This summing is indi- 
cated by the so-called sign of integra- 
tion /, which is placed before the general expression of the differ- 
ential to be summed. Thus we write, instead of 

y = [^ {dx) + (f) {2dx) + (f) (3 d x) + . . . + </> {X)] d x, 
y — f (p{x) dx. 

In this case we call y the integral of {x) d x^ and (j) (x) d x the 
differential of y. Sometimes we can obtain the integral / </> (x) d x, 
by really summing up the series {d x), (j) {% d x), (f) (3 d x), etc. ; 
but it is always simpler in the determination of an integral to em- 
ploy one of the Kules of what is known as the Integral Calculus, 
which will be the next subject treated. 

If n is the number of differentials dx oix, we have x = n d x 

X 

OT d X = -, and we can put 

Thus for the differential d y = a x d x^we have 
y = /axdx = adx(dx-{-2dx-{-3dx+,,. + ndx) 
= (1 + 2 + 3 -\- . . . + n) a d x% 
or since according to Art. 15, IV., for ^ = oo we have the sum of 
the natural series of numbers 

^ fi z=z ^ n^ and d x^ -— 



X 

n 



1 + 2 + 3 + 4 + 5 



n 



y = faxdx = hn^ 



a 



X' 



= ^ ax^ 



In a similar way we find, if x = n d x or if xis composed of n 
elements d x, 

yz=zf<^{x)dx=f—^\{dxY-\-{%dxY^{ZdxY^,,,-\-{ndx)^ — 

a \- -J a 

= (1 + 2=' + 3' + .... + ^^)—. 
^ ^ a 



Art. 17.] INTRODUCTION TO THE CALCULUS. 61 

But from § 15, IV., forn = co , we have 

1 4_ 2' 4- 3' + . . . . + w'' = -^j whence it follows that 

/x^ dx _n^ d x^ _ (n d xY _ ^ 
~~a^ ~~ d ' a ~ Sa ~ 3a 

Akt. 17. From the formula d (a -\- mf{x)) = m df(x), we 
obtain by inversion 

fmdf (x) = a + m f (x) = a + mf df (x), or putting 
df (x) = (p (x) . d X 
I.) f m4> {x) dx — a + m f (p {x) dx, 

and hence it follows that the constant factor m remains, in the In^ 
tegration as in the Differentiation, unchanged, and that a constant 
member such as a can not be determined by mere integration; 
the integration furnishes only an indefinite integral. 

In order to find the constant member, a pair of corresponding 
values of x and y =.f(p(x)dx must be known. If iorx = c^y = k, 
and we have found y — f (p (x) d x = a + f (x) then we must 
also have h = a + / (c), and by subtraction we obtain y ^ h = 
f (x) —f (c) ; therefore in this case we have 

y = f<p{x)dx = h +f{x) ~f(c)=f(x) +h-f(c), 

and the constant factor a — h — f (c). 

When, E.G., we know that the indefinite integral y = f x d x — 

x^ 

-^ gives, for a; = 1, y = 3 we have the necessary constant a = 

3 — ^ = 1, and therefore the integral 

/-. X O ~\~ X 

xdx = a -\- Y ~ — 2 — * 

Even the determination of the constant leaves the integral still 
indefinite, for we can assume any value for the independent varia- 
ble x ; but if we wish to have the definite value hi of the integral 
corresponding to the definite value Ci of x, we must substitute this 
value in the integral which we have found, or, Tcx^=h-\-f {Ci)—f(c). 

E.G., y — f X dx — — - — gives, for a; = 5, y = 15. 

Generally the value of x for which y becomes = is known ; 
in this case we have ^ = 0, and the indefinite integral of the form 



62 INTRODUCTION TO THE CALCULUS. [Art. 18. 

f <t>^ {dx) ^f{x) leads to the definite one ki = f (c,) — f {c)y 
which can alco be found by substituting in the expression / (a;) 
of the indefinite integral the two given limits, Cx and c, of x, and 
by subtracting the values found from one another. In order to 

\ndicate this we write instead ot f ((> {x) d x, I {x) d x, 

if, ^.G., / cl> dx= J, Jpcp{x)dx = -^-^. 

By the inversion of the differential formula 

d [f (^) + (^)] = df (^) + d (j) (x) we obtain the integral 
formula/ [<^/ (x) + d (p {x)] — f {x) -f {x\ or putting 
df {x) ~ ip (x) d X and d (p {x) = % (^) d x, 

11.) f [i^ (x) dx -^ X (^} dx] = f^ (x) dx -h f X {^) d X. 

Therefore the integral of the sum of several differentials is equal 
to the sum of the integrals of each of the differentials. 

E.G. /(3 + bx)dx — f^dx-\-fbxdxr=^Zx + ^^x\ 

Aet. ] 8 . The most important differential formula, lY., Art. 8^ 
d {x'') — . n af~^ d X, gives by inversion an integral formula which 
is equally important. 

Itis f n a;""^ d x = x", or n f x"'^ d x = x", whence 



iC" 



f x"~^ d X — — ; 
n 



substituting *i — 1 = m, and n = m -h l,we obtain the following 
important integral : 



/ x'^dx ^ 



/j,TO+l 



m + V 

which is employed at least as often as all the other formulas 
together. 

The form of this integral shows that it corresponds to the sys- 
tem of curves treated in Art. 9 and represented in Fig. 17. 

From it we have, e.g., 

f 6 x' d X =: 5 f x^ d X = ^ X* ; 

f y^d x=f x^dx = -I xl = f V^; 
f{4:-6x''-\-6x*)dx = f4:dx~f6x'dx-^f5x'dx 
= 4:fdx- efx'dx + 6fx*dx=4:X-2x' + x'',Mt'heT, 

putting Z x — % = u, 3dx = du, ovdx = -^, we have 



Art. 19.] INTRODUCTION TO THE CALCULUS. 63 

and finally, substituting 2 x" — 1 — u and 4: x d x = d i\. ot 

, d u ^ 
X d X = -j-, we nave 



4 2 
3 



P 6 X dx /^b du = /* _i -, 

o o 

By the substitution of the limits the indefinite integral can be 
changed into a definite one. 



.G.J^' 6x'dx = i{2' -r) = i.{16 -1) = 18|. 

/ 



'^ d X ^ ^ 

-— :^ 4/9 _ 1/4 =.1 

4 2 y ic 



^ VZx-2.dx^\{ 4/16^ - 4/1^) = I (64 - 1) = 14. 

If E.G. /(4 - 6 a;' + 5 ic') ^ .7; = 7, for a; = we would have, 
in general, 

/ (4 - 6 ic' + 5 a;^) 6? a: = 7 + 4 a; - 2 a;^ + a;^ 

Art. 19. The so-called exponential function y = a"", which 
consists of a power with a variable exponent, can be developed as 
follows into a series by means of McLaurin's Theorem, and its dif- 
ferential can then be found. 

Putting a' = Ao + A^ X -h Ai c(f -\- A^ x^ + . . . , we have, for 
fc — Oy a' = a"" = 1^ whence Aq = 1; 

From a'' = 1 + A^ x + A^ x"" + A3 x^ + . . . . we have 

a*^* = 1 + A^ d X + Aa d x"" + Azdx"" + and also 

d {a') = a'+^' — a' = a" of' — a^ z=i a" (a''^ — 1) 
= a'' {Ai d X + A<id x^ + A^ d x^ + ....) 
= a' {Ai -{- A^d x + . . . ,) d X = A-i a" d x. 

Hence, by successive differentiation of the series, we have 

f {x) = a' = 1 •\- A^x + A.iX' + AzX^ + . , , 

f, {x) = ^ = A,a' = A^ -\- 2A,x + SA,x' + ... 

Cl X 



64 INTRODUCTION TO THE CALCULUS. [Abt. 19, 

fi (^) = ^-^^~^ =A^a' = 2,3.A, + ... 
Putting a; = 0, it follows that 

whence ^= = ^^^ ^.S A. - j-^ ^,S ^, = JT^^^A &c. 
and the exponential series takes the form 

The constant coefficient ^i is of course a definite function of 
the constant base, as the latter is a function of the former. If one 
of the two numbers be given, the other is then determined. The 
most simple, or the so-called natural series of powers, whose base 
(«) will be designated hereafter by e, is obtained by putting ^i = 1. 
Then we have, 

IL) 6=1 + ^ + ^-^ + j-y^ 4- 1 2.3.4 "^ • • • 

and if we put a; = 1 we obtain the lase of the natural series of 
powers, 

e' = e = 1 + 1 + 1 H- I + 3^L + = 2,7182828. 

1 1 

If we put e = (Q^"*, or « = &", we have — = la, which is the Nape- 

lit 

rian or hyperbolic Logarithm of a, and 

III.) or = (rY ^ e^ = 1 + -{-] ^ j-Q ( ~) + 

17273 W '^••* 
Since this series corresponds in its form to that of I, we have 

also A^ = — , and, 
m 

Oj^ d X 
IV.) d (a") = Aitt' d X = = I a . a' d x, as well as 

V.) d {e') = e'' dx. 
E.G. d {^'-^') = 6'"+' ^ (3 ic + 1) = 3 e^^+' d x. 



Art. 30.] INTRODUCTION TO THE CALCULUS. 65 



K we put y ~ a'' z=z e"* we have, on the contrary, 

X = log, «/ and — = I y. 

log,, y = ml y, and, on the contrary, 

I y, or log, y = ~ log, y. 

The number m is called the modulus of the system correspond- 
ing to the base a. By means of it we can transform the Naperian 
logarithm into any artificial one, or one of the latter into the 
former. For Brigg's system of Logarithms the base is « = 10, 

whence — = ? 10 = 2,30258, and, on the contrary, m = ttt^ = 

0,43429. 

We have also log y = 0,43429 I y, and 
I y = 2,30258 log y. 
(See Ingenieur, page 81, etc.) 

Akt. 30. The course of the curves which correspond to the 
exponential functions y — e", and y = lO'', is represented by Fig. 
32. For a; = 0, we have in both cases ^ = e° = a" = 1. Hence 
both curves Q 8 and Qi Si pass through the same point ( 0) 
of the axis of ordinates A Y, For a? = 1 we have, 

y = e - 2,718, and 
y =Vy> = 10, 
a; = 2 gives 

y = e=. 2,718^ = 7,389, and 
2^ = 10^ = 10^ = 100, &c. 

Both curves rise on the positive side of the axis of abscissas very 
steeply, particularly the latter. 

For a; = — 1 we have e =^ e-' = ^^—r^ = 0,368 . . , and 

10^ = 10-^ = 0,1; 
farther, for a; = — 2, we have 

and 10' = 10-=^ = 0,01; 

for a; = — oo both equations give 
5 



66 



INTRODUCTION TO THE CALCULUS. 



[Art. 21. 




The two curves approach 
nearer and nearer this axis of 
abscissas on the negative side 
of the axis of abscissas, the 
last more quickly than the first, 
but they never really meet thia 
axis. 

Since we deduce from the 
equation 

y zzz e^, X = I y 
and also from 

y =z a%x = log„ y 
the abscissas of these curves 
furnish a scale for the Nape- 
rian and common logarithms ; 
for the abscissas are the loga- 
rithms of the ordinates. 
E.G. we have, 
A M= IMF 

= log„ M Pi, etc. 
From the differential for- 
mula IV of the last article the 
tangential angle of the expo- 
nential curve is determined by 
the simple formula, 

_ d y a" d X 
~ d X 
a" 
m 



tang, a 



m d X 
y la. 



Consequently for the curve 



y - 

m 
Pi Qi Si, Fig. 32, the subtan- 



gent = y cotg. a 



m. 



that is, is constant; and for the curve 
P Q S it \^ always = 1, E.G., for the point ft ^ 1, = 1 for the 
point i?, 12 = 1, etc. 

Art. 21. If a; = a\ we have also 



and by inversion, 



d X 



dy 



T , . a^ d u 



m 



m d X m dx 



cC' 



X 



Akt. 22.] INTRODUCTION TO THE CALCULUS. 67 

But y — log a ^, that is, to the logarithm of the variable power 
X with the constant base a ; therefore we have the following differ- 
ential formula for the logarithmic functions, 

y ~ log^ X and y = I x: 

■y S^ -1/1 \ ''" (a/ X \. Cv X 

I.) d(log^x)^^^ = :^^-—, 
II.) d(lx)= —. 

X 

If a is the tangential angle of the curve corresponding to the equa- 

TYl 

tion y = log a x, we have tang, a = — , and the subtangent = y 

X 

cotg. a = -^, or proportional to the area x y of the rectangle con- 
structed with the sides x and y. 

By means of the differential formulas I. and II. we obtain 



1) '?(?^*)=^7^ = 



2 /• 

d^x_d{xl)_^ x~^ d X _ d X 



iCf '-^ x\ 2 X 



or also 



=z d (^Ix) — ^ d (Ix) = ^ . - — . 
H) dl^^-^ = d[l{2 i- x) - Ix'] 

= dl {2 -h x) — dl (x') 

_ dx i)<^ X _ {4: + x) d X 



2 + X X X {2 -\- x)' 

3) d (l '^\) = d [?(.- -!)]-<? {l(e' + 1)] 

_ d (f) d {e') _ e" d x e" d x _2 e' d x 

~ e"—! ~ "e^+T ~ e^-^ ~ e' + 1 ~ e^' — T 

Art. 22. If we reverse the differential formulas of the fore- 
going article, we obtain the following important integral formulas. 

af" d X (^Oj^ d X 

From d (a') = it follows that / = a% I.E., 

m ^' m 

I.) f a" dx = m a" = a" \l a, and therefore 
II.) f e^ dx = e\ 

Farther, from d{logaX)=- , it follows that / = %« x, lb. 

X ^ X 



68 INTRODUCTION TO THE CALCULUS. [Art. 23. 

/d X 1 
— ■ =■ — loQa X — IXf which is also given by the for- 



X m 
mula d {I x) — — . 

X 



By their aid we can easily calculate the following examples: 
f e'^' dx = If &'-' d (5 a; - 1) == i &^\ 

/( J^-t) '^ ^ = /(^ + 1 + ^1^)^ ^■ 

= fxix + fdx + % f^Jl^i) ^ ^ + a, + 2 ? (a; - 1). 

x^dx — leayes 

m + 1 

the last integral undetermined ; for putting m = — 1, it follows 

/d X P x^ 

— = / x~^ d X = ~ + SL constant = 00+ constant, but 
X ^ 

if we put a; = 1 + ^^, and d x = d u, wq have 

fi X f] 1/ 

-^— — — (1 — u + u^ — u^ + zt* —....) du; and therefore 

a; 1 + ^^ . 

— —I ^ = (1 — u -\- u" — u^ -\- u^ — ) dti 

X ^ 1 + u J 

= fdu — fudu + f u^ d u~- f u^ du + .... 



w u^ W' 



-^-^ + -^ 4-+----; 



u"^ w w 



we can therefore also put l{^-^u) — u ^ 4- -^ ^7 + • • v or 

IV.) lx={x-V) - ^ ^ ^ + ^ g ^ - -—4—^ + • • •• 

With the aid of this series we can calculate the logarithm of all 
numbers which differ very little from 1 ; but if we require the 
logarithm of large nun/jers we must adopt the following method. 

Taking u negative in the foregoing formula, we have 

■t i-t \ lb u u 

l{\-u) = - ic - - - — ~ "4 ^ ' • •? 
And subtracting one series from the other, we have 



Aet. 83.] INTEODUCTION TO THE CALCULUS. 69 

l(l + u) -1(1- u) ^ 2 (u +j + ^ + ...) 

\x~J = 2 (« + 3- + 5 + • • •) or putting 

1 + tc X — 1 

j—^^^, or 2. = ^^, we have 

^ L:?; + 1 ^ Ma: + 1/ ^ M:?; + 1/ ^ J 

This formula is to be employed for the determination of the 
logarithm of such numbers as differ sensibly from 1, since 

X — 1 . 

' — -^ ^^ always less than 1. 

X ~t~ A. 

We have aho I {x + y) - I x = I (^-^) = ^^ + ^ 

=^_,(y)V.(yy_etc. 

X '^ \xl ^ \xl 



-^L2^n/+n2x^ + J + n2^ + J "^•••J 



whence 



.2x + y ^\%x-\-yl ^\2x + y) 

VI.) l{x -{- y) = lx + 2\^^— + i (^r^— )V • . .] 

♦ 
This formula is used to calculate from one logarithm, that 
of a somewhat greater number 

E.G., l2 = 2[-^l-+i.(l^)\...] 

/ 0,33333 \ 

= 2 ^'^1^^^ [ = 2. 0,34656 = 0,69312, 
] 0,00082 ( 

( 0,00007 ) 

more exactly = 0,69314718. 

Hence / 8 = 12' = S12 = 2,0794415, and according to the last 
formula, nO = / (8 + 2) 

= 2,0794415 + 0,2231436 = 2,302585. 



70 



INTRODUCTION TO THE CALCULUS. 



[Am:. 24. 



Wecanalsoput ?2 = Z1 + 2[^-^ + J U^J^ ••••] 



= 2 U -4- J i- + 1 -i- 4. 



3 -T- 3« 33 -1- 5--3^ -T- •••• ) = 0,693147; 



father, I6 = l{4. + l) = 2l2-h 2(1 + ^-^ + ....), and Anally 

we Ciin put / 10 = ?2 + Z5. 

(Compare Art. 19.) 

Aet. 34. The trigonometrical and circular functions, whose 
differentials will now be determined, are of practical importance. 
The function of the sine,^ = sin. 2;, gives for :r = 0, ?/ — ; 

IT 3,1416 



for x = 



= 0,7854..., y =1/1 = 0,7071, 



77 



" X = -^ 7T,y = — 1, for X = 2 rr, y = 0, etc. 
Taking x as tke abscissa A 0, and y as the corresponding ordi- 
nate P, we obtain the serpentine curve {AP Bn G2n), Fig. 33, 
which continues to infinity on both sides of A, 



-X 



T K E 



Fig. 33. 
M 



Cr 





+ 


1 \ 




1 


4 


1 1 

1 1 


h 


j\^ 


/ 


A ( 


' ^iV 


y^Nv 


/l\ 


y2i;r 1 


If 


^..., 


_. 


ti 


z.__ ^__ 


:c"""7 


/_ — 1 





X 



-Y 



F 



H 



]sr 



Akt. 24.] INTRODUCTION TO THE CALCULUS. 71 

The function of the cosine,?/ = cos. i», gives, ioT x = 0,y = 1\ 
for x= -r,y= V^'y ioix = —,y=0'y for x = tt, y = — 1 ; for 
X = ^ n, y = ; for x = 2 7T,y — 1, etc. ; it corresponds to exactly 
the same serpentine line ( + 1 P — Z) — + 1 j as the function of 

the sine, but it is always a distance i tt = 1,5708 behind or in 
front of the curve of the sine. 

The curves, corresponding to the function of the tangent or co- 
tangent,?/ = tang, a and y — cotang. x, are, however, of an entirely 
different form. 

If we substitute in. y = tang, ic, a; = 0, | tt, J rr, we obtain y = 0, 
1, 00 , and therefore a curve {A Q E) which approaches more and 
more, without ever attaining it, a line parallel ^o the axis of ordi- 

nates A Y, and cutting the axis of abscissas ^ X at a distance - 

from the origin of the co-ordinates. Now if we put Xr = - , n, | tt, 

we obtain ?/ = — oo, 0, + c», and therefore a curve {Fn 0), which 

continually approaches the parallel lines, passing through I - I and 

(I tt), and for which these parallel lines are asymptotes. (See 
Art. 11.) 

If we increase x still more, the same values of y are repeated, 
and therefore the function y = tang, x corresponds to a series of 
curves which are separated from each other in the direction of the 
axis of abscissas by a distance tt = 3,1416. On the contrary, the 

TT TT 

function y = cot. x gives for x = 0, - -^n, y = oo, 1, 0, — oo, and 

therefore corresponds to a curve IKQ- Lj which differs from the 
tangential curve only by its position; it is also easy to perceive 

correspond to this function. 

While the curve of the Sine and Cosine forms a continuous, 
unbroken whole, the curve of the Tangent as well as that of the 
Cotangent is formed of separate branches ; for the ordmates for 
certain values of x change from positive to negative infinity, in 
consequence of which the curve naturally loses its continuity. 



72 



INTRODUCTION TO THE CALCULUS. 



[Akt. 25. 



Akt. 95. The differentials of the trigonometrical lines or 
functions are given by the consideration of Fig. 34, in which 

G A = C P = G Q = l,2LrcA P = x,P Q = dx, 

P M = sin. X, G M ~ cos. x, A S = tang, x, 

OQ^NQ — MP=. sm. {x -{- d x) ~ sin. x — d sm. x, 

P =■ — {G N— G M) = — COS. (x + dx)-}- cos. x —— d cos. x, and 

S T = A T — A S = tang, {x + d x) — tang, x — d tang. x. 

Since the elementary arc P Qh perpendicular to the radius 
G P, and since the angle PGA between the two lines G P and 
G Aw> equal to the angle P Q between the two perpendicular to 
them, P Q and Q, the triangles G P M and § P are similar, 
and we have 



Q G M d sm. x cos. x 



I.E. 



P Q G F "^' dx 1 

I.) d {sin. x) = COS. x . d x, and in like manner, 



whence 



OP P M 



11.) d {cos, x) = 



I.E. 



— d COS. X sin. x 



dx 



1 



, whence 



Fig. 34. 



G P' 

- sm. X d X. 

We see from this, that the influence of 
errors in the arc or angle upon the sine 
increases as cos. x becomes greater, or as 
the arc or angle becomes smaller, while on 
the contrary their influence upon the co- 
sine increases as sine x becomes greater, 

that is, the more the arc approaches to - , 

/* 

and that finally the differential of the co- 
sine has the opposite sign from that of the 
arc, for we know that an increase of x 
causes a decrease of cos. x, and a decrease 
of X an increase of cos. x. 

Letting fall a perpendicular 8 R upon 
G Twe form a triangle SET which is 

similar to the triangle G P M, since the angle R T S is equal to 

G Q ]V or G P M, and we have 




N M 



8_T 
SR 
S R 



GP 
G M' 

PQ 



I.E. 



d tang, x 



G 8 G P 



, I.E. 8R = 



8 R cos. X 

G 8 .dx 



but we have also 



and 



ART. 26.] INTRODUCTION TO THE CALCULUS. 73 

1 dx 
C S = secant, x — , whence 8 R — and 

COS. X COS. X 

III.) a (tang. X) = ^^,. 

If instead of x we substitute ^ — x, and instead of dx,dl- —x\ 

= — d x,we obtain 

d tang. [^ - ^) = - = j„ rr,, i-E., 

\cos. (^ - x)\ 

IV.) d (cotanq. x) — — -^^ tt,. 

' ^ ^ ' \s%n. X) 

By inversion this formula gives for the differential of the arc 

, d sin. X d cos. x : x- , ^ 

d X = • = . = (COS. X) d tana, x 

COS. X , sm. X ^ 

= — (sm. xy d cotang. x, or 

, d sin. X d tana, x ,. 

dx = - ^ = -z — . ,, ^ — ; 2, as well as 

VT^(sin. xf 1 + {i(^^ff- ^) 
^ /7 — ^ ^^^' ^ — __ ^ cotang. x 

' ~ 4^1 - {cos. xf ~ 1 + (cotang. xf 
If we designate sm. x by y, and x by sinr^ y, we have 

V.) dsin.~^ y — 



Vl-f 
and in the same manner we find 

VI.) d cos.-' y=- ^y 



VI - / 
VII.) d tang.-' y = ^-^ 

VIII.) d cotang.-' y — — ^ g . 

Art. 26. By inversion the latter differential formulae give 
I.) • f COS. X d X ^=^ sin. x, 
II.) / sin. X d X ^^ — COS. x, 



IV.) / -7— r — = — cotang. x, 



* /Sen.-^ y, tang.-^ y, etc., designate the arc whose sine is y, whose tangent 
is y, etc. — Tr. 



74 mTRODUCTION TO THE CALCULUS. [Abt. 26. 

V.) / ^ = sinr^ x = — cosr' x, and 

"^ Vl - x' 

VL) / = tangr^ x — — cotangr^ x. 

^ Vl + x^ 

iTom the aboye, since we have a (I sm. x)= — ■. = - — r- 

sin. X sm, X 

= cotg. X . d X, we can easily deduce 

VII.) / cotg. X d X = I sin. x, and also 

YIII.) / tang, x d x = — I cos. x ; further 

d tang, x _ d x _ d x _ d (2 x) 



d (l tang, x) 



tang, x cos. x^ tang, x sin. x cos. x sin. 2 x' 



whence d (I tang, i x) = — — , and 

SDl. X 

^^.1 ah a{l—x) + 'b{l^x) 

Now putting — r = + :r~ — = —7^ x ,-. r^, 

^ ^ 1 — x^ 1 + X 1 — X {1 -hx) (1 — x) ' 

we have 1 = a{l — x) -\- h (1 + x), and taking 1 4 i?^ = 0, or x = 

— 1, we obtain 1 = a{l +1) whence a = i, and putting 1 — a: — 0, 

or a; = 1, we obtain 1 = 2 J or 5 = ^, whence 



1 



+ -—^ — ; and finally 



1— x'^ 1 -h X 1 — X 

/iS-^=i/i^.+i/i^ = 1 m + .) - i ? (1 - .), I.E., 

XL) / z ^ = ^ t (- ), and in like manner 

' ^' 1 — x^ ^ M — x/ 



Art. 37.] INTRODUCTION TO THE CALCULUS. 75 



Putting Vl-{- x" — xi/,we have 1 + x"" = x^ y' and 
dx(l — y^) = xy dy, whence 



f5_ _ dy _i^. J (i+J^ 
Vl + x'~ 1- f~-' \l-yJ 



XIIL) f y^ — , =l{x+ Vl+ of), and also 

Art. 27. In order to find the integral of tangr^ x= I — 



d X 



a 



we have only to change into a series, by division, and then 

J. "T" X 

integrate each member. We obtain thus 
1 



1 + x' 



x'' -\- X* — x^ + x^ — . .., and 



/ -^= j dx— I x'^dx-\- I x^dx— j x^dx-\-,,., consequently 

/y>^ /y*^ /y** 

I.) tangr^ x — x — '^ ^ 't i^ — ••• ©^c., e.g., 



— = tangr^ 1 — 1 — \ 4- i— 4 + i~---j ^^<i ^'^ half circumference 
7r=:4(l-| + i-4 + ^-...),or 

whence 7r= 6 i^f (1 - i + 4V - 759 +-..) = 3,1415926.... 
In the same manner we obtain from 

' y'-j __ 2 = (1 — ^') ^ = 1 + ^ a:; + I a? 4- jg a; + . . . 
j ——^===1=1 dx-\-^ I x" dx-^lj x'' dx + ^^J x^dx+,,,, l.E.> 



ic 



-|-T \ » 1 -L 2/ A. t d X J..O.02/ 

11.) si^. ^ = ^+273 + 2:47^ + 2.4.6.7"^***' 

E.G., — = Sin.~ 5 = 2 (I + 2V + 54 + 71^5 + • • • )> 



76 INTRODUCTION TO THE CALCULUS. [Art. 28. 

1,04167 

^=3. /^'^^^^^ =3,1416.. 
^ 0,00070 ' ' 

0,00012 

When we put sin. x = Aq + AiX + A^x" + A^x^ + A^x* + •••^ 
etc., we obtain by successive differentiation 
d (sin. x) 

cl X 
d {cos. x) 



d X 



— cos.x= Ai + %A^x + ^ A^x" + 4:^4^2;^ + ... 
— sin.x = 2^2 + 2.3^3a; + 3. 4-44^' +... 



d (sm. x) ^ ^ J ^ ^ . J 

^ ^= — C05. a; = 2.3.^3 + 2.3.4.^4 0; + ... 

ax 

d (cos. x) . ^ ^ ^ , 

J ~ = sm, a; = 2.3.4.^4 + .,. 

d X 

Now for x = we have sin. x = 0, and cos. x — 1, therefore 
we obtain from the first series ^o = 0, from the second Ai = cos. 

= 1, from the third ^2=0, from the fourth Jg = — - — -, from 

/i . 3 

the fifth A4 = 0, etc. If we substitute these values in the supposed 
series, we have the series of the sine 

.) sin.xz=-.--^^ + 



1 1.2.3 1.2.3.4.5 1.2.3.4.5.6.7 
In the same way we obtain 



x^ X* X" 



IV.) COS. x-l-^—^ + 1.2.3.4 -1.2.3. 47576 ^ '"' 

V\ , X /ii X J. I X 1 

.) tang. x = x + -^ + ~ — - + „ ., ^ „ + — and 

o 0.5 0.5.7.0 

VI.) '">i''«ff-^=l-l-37h-3-^7r:9-'^^- 

(See Ingenieur, page 159.) 

Art. 28. When we integrate the differential formula d (u v) 
:=udv-\-vdUjOf Art. 8, we obtain the expression uv ~f udv 
-{■/ V d Uy and the following formula for integration : 

fvdu^=uv— fudv,OY 

f (P {X) df {X) = (/) (x) f {x)-ff (x) d (x). 

This is known as the integration by parts. 

This rule is always employed if the integral / v d u = 
f ^ {x) df (x) is not known, and if, on the contrary, f u d v— 



Art. 28.] INTRODUCTION TO THE CALCULUS. 77 

f f {x) d <() X is. E.G. By means of this formula we can refer the 
integration of the formula, 



d y = Vl + x" , d X 
to another known integral. We must substitute 



</) {x) = VI + x\ whence d (t> (x) = y^-^^, 
and / (x) = Xf whence df (x) = d x, then we have, 

fvTn^dx = xVTT^^ - f-^f=.' ^^* 

x"" 1 + x'' 1 



= Vl +x'- 



Vl+X' Vl+X"" V\^X^ ^/\^\^ X'' 

whence it follows that 

f VlTx' dx = x VlT~x' - y* VTT~x' dx rh y*- ^^-— ^, > ^^ 

/> d X 

and consequently, 

I.) f^^rv^'dx = ^xViT^^ + if^^^~, 



= i[x vr+x' -h i(x -h vi + x')\ 

In like manner, 

n.) fvr:r^dx = i,xVr:^-' + if^^== 

z= ^\x Vl — x"^ + sinr^ x], and 
III.) f V^?^ldx = ^,xV^^l-^,f-^^=^ 



= H^ ^^^^^ -l{x + V^'- 1)]. 

We have also 

/{sin, zy d x=fsin. x sin, x dx= —/sin. x d (cos. x) = —sin, x cos, x 
-{-/cos. X d {sin. x) = — sin. x cos. x -{-/ {cos. xY dx 
= — sin, X cos, X + /[I — {sin. xY] d Xy 

whence it follows that 



78 



INTRODUCTION TO THE CALCULUS. 



[Abt. 29. 



2 / (sin. xY d x z=:f d x — sin, x cos, x, and 
1^0 / (^^'^' xy dx — \{x — sin. x cos, ^) = I (x — ^ sin, 2 x). 
In like manner 

V.) / (cos. xf d x=-^ (x + sin. x cos. x) =^ (x + ^ sin. 2 x), and. 

VI.) / sin. X COS. X dx = \ f sin. % x d (% x) — — \ cos. 2 x, 

Vll.) / (tang, xf dx — tang, x — x, and 

VIII.) / (cotg. xy d X = — (cotg. x + x). 

Finally we have 

IX.) / x sin. X d x= —x cos. x + f cos. xd x= —x cos. x + sin. Xy 
X.) fxe'dx^fxd (e") — x e" — f e'' d x — (x — 1) e', 



.) Ilx.dx = xlx— Ix — = X (I X — 1), and 



XII 



«// (Jj JU 



X' 



■•) I X V X . a X — -zr~ i X — / ~^ — \v X ~~~ TV ) _ 
^ t/ 2 ty 2 X ^ ^^2 



Fig. 35. 




Art. 39. If we wish to find the quadrature of a curve, A P JB^ 

Fig. 35, I.E., to determine or express by 
a function of the abscissas o this curve 
the area of the surface ABC, which 
is enclosed by the curve A P B and 
its co-ordinates A G and B (7, we im- 
agine this surface divided by an in- 
finite number of ordinate s M P, N Q, 
etc., into elementary strips, like M N 
P Q, with the constant width d x, and 
the variable length M P — y. Since 
f^e can put the area of such ail element of the surface 

dF =^ ( ^^^^^ ) ,MN=(y + }sdy)dx = ydx 

we will find the area of the entire surface by integrating the differ- 
ential y d X, and we have 

F = f y dx\ 

E.G., for the parabola whose parameter is p we have y"^ = p x,- and, 
therefore, its surface 



MjV 



F = I Vp xd X = Vp I x-i d X = — ^ — = ^ xVp X = \ 



xy. 



Art. 29.] 



INTRODUCTION TO THE CALCULUS. 



79 



The surface of the parabola A B Cis therefore two-thirds of 
the rectangle A G B D which encloses it. 

This formula holds good also for oblique co-ordinates inclined 
at an angle X A Y = a, e.g., for the surface ABC, Fig. 36, we 
have when we substitute instead of B C —y the normal distance 
B N= y sin. a 

F = sin. a f y d X, 

E.G., for the parabola when the axis of abscissas J[ Xis a diameter, 
and the axis of ordinates J^ 1^ is tangent to the curve, we have 

y"^ = p^x = ^?-Y-- (See "Ingenieur," page 177.) 
and F = ^ X y sin. a, 
I.E., the surface A B = ^ parallelogram A B C D. 




Fig. 37. 




For a surface B C Ci B^ = F, between the abscissa AG^ = (?» 
and A G — c, Fig. 37, we obtain, according to Art. 17, 



F 






a^ 



E.G., for y = ~, 



X 



F 



I.E., F 



"1 a^ d X 






a^ {I Ci — I c), 



a 



The equation - corresponds to the curve P Q, Fig. 38, dig- 

cussed in Art. 3, and if we have A M = c and A N = c,y the area 
of the surface M N Q P \^ 



80 



INTRODUCTION TO THE CALCULUS. 



[Art. 30. 



F 



= «'K3 



Fig. 38. If we suppose, for simplicity, 

that 6? = c = 1, and Cx = x, we 
obtain 

F =lx\ 

hence the surfaces (1 MP 1), 
(1 iV § 1), etc., are the Naperian 
logarithms of the abscissas A M, 
A N, etc. The curve itself is the 
so-called equilateral hyperiola in 
which the two semi-axes a and h 
are equal ; hence the angle formed 
by the asymptotes with the axes is a = 45° ; and the right lines 
A X and A Y, which approach nearer and nearer the curve with- 
out ever attaining it, are its asymptotes. In consequence of the 
relation between the abscissas and the area of the surfaces, the 
Naperian logarithms are often styled hyperbolic logarithms. 




— X 



Fig. 39. 



B 



AeT. 30. We can put every integral fydx = f(^{x)dx 
equal to the area of a surface F, and if the inte- 
gration cannot be effected by means of one of the 
known rules, we can find it, at least approximately, 
by calculating the area of the corresponding 
surface by means of a well-known geometrical 
device. 

If a surface A B P Q N, Fig. 39, is deter- 
mined by the base A N = x, and by three equi- 
distant ordinates A B — y^, M P = y^, N Q 
= y^, we have the area of the trapezoid 



Q 



R 



M 



N 



X 



ABQN=F, = {y,^y,)-', 

and that of the segment B P Q S B, if we consider B P Q 
to be a parabola 



F, = iP S.BR = i {MP-MS).AN=lU- 
Hence the entire surface is 



^0 + y^ 

2 



\ X. 



Art. 30.] INTRODUCTION TO THE CALCULUS. 



81 



X 



X 



= [h (yo + y^) +iyi]^ = {yo -^^yi + y^)--^- 



If we introduce in the equation a mean ordinate y and put 
F=x y,we obtain 



y= 



3/0 + 4^1 + y, 
6 



In order to find the area of a surface, lying above a given base 
M N = X, and determined by an uneven number of ordinates 
yoi y\i «/25 2/3 •• • Vm by which it is divided into an even number 
of equally wide strips, we have only to make repeated application 



X 



of this rule. The width of a strip is -, and the area of the first 

n 



Fig. 40. 



pair of strips is 

_ 2/0 + 4 2/1 + 2/2 2 iz; 




M 



« > 



.■..N^vw».>a&s^N 



n 



B 6 

of the second pair 

_y, + 4:ys -h y^. 2x 



6 
of the third pair, 



n 



N 






n 



and the area of the first six strips, or of the first three pair, for 
which n = 6,is 

X 

F= {yo + 4. y^ + 2 y, + 4: ys + 2 y, -\- 4:ys + y^)^^ 



X 



Oo + 2/6 + 4 (2/1 + y,-hys) + 2 (y, + y,)] ^; 

it is easy to perceive that the area of a surface divided in four pair 
of strips is 

X 

F= [2/0 + 2/8 + 4(2/1 + 2/3 + 2/5 + yt) + ^ (2/2 + 2/4 + 2/6)] 37g» 

and in general, for a surface divided in n strips, we have 

X 

^=[2/0+2/n+4(2/l+2/3+- + 2/n-l) + ^ (2/2 + 2/4 + - +2/n-0] g"^* 



82 INTRODUCTION TO THE CALCULUS. [Aet. 30. 

and the mean altitude of such a surface is 

,.^ y^ + yn + ^ {yi + yz + ♦'■ + y„-i) + 2 {y, +3/4 + .>■ + Vn-,) 

in which n must be an even number. 

This formula, well known under the name of Simpson's Rule 
(see "Ingenieur," page 190), can be employed for the determina- 
tion of an integral / y dx— I (ic) ^ ^, if we divide x^ Cx — c 
into an even number n of equal parts, and calculate the ordinates, 
y,=(i> (c),y, = (l>{c + ^,y, = (i>\c + -^j, 

«/3 = (c + 1 ... up to ^„ — {x), 

and then substitute these values in the formula 



/ y d x= I (l){x)dx 

= [2/0 + yn+^ {yi + 2^3 + .. + yn-x) + 2 {y^+ 2/4+..+ yn-^) ] y^ 

P'^ dx 1 

E.G., / — gives, since here Ci— c=2— 1=1 and y=^ {x) = -, 



when we assume w = 6 or - = ^-77— = J, 



it- Cj C 

n 6 



y, == A := 1,0000, y, = i = f = 0,8571, I/, =^ = I = 0,7500, 

-L 6 6 

1 1 <^ 

2^3 =: -=::« =0,6666, 2/4=7^=^0,6000, ^5=--=0,5454, and ^6=^0,5000, 

6 6 , , ■ 

therefore 

y^ -f 2/6 = 1,5000, y,^y^ + y, = 2,0692, and y^ + y^ = 1,3500, 
and we have the required integral 

£ ^/^r= (1,5000+4. 2,0692+2 . 1,3500) . yV=?^^^= 0,69315. 

From Art. 22, III, we have 

/'*— = Z 2 - a = 0,693147. 

«/i X 

We see that the results of the two methods agree very well. 



AnT. 31.] 



INTRODUCTION TO THE CALCULUS. 



83 



Aet. 31. Further on, another rule will be given which can be 

employed for an uneven number 
of strips. If we treat a very flat 
segment A M B, Fig 41, as a seg- 
ment of a parabola, we have from 
Art. 29 the area of the same, 

F= ^ A B .MD, 
or, if A T and B T are the tan- 
gents at the ends A and B, and therefore C T — "^ G M^^q have 

i^ = I . ^ = I of the isosceles triangle A S B of the same 




CDE 



height, and therefore = ^AC.CS— ^AC^ tang. SAC. 

The angle SAC = SBCis = TAC+TAS=TBC- 
T B 8', putting the small angles T A /S'and T B S, equal to each 
other, we obtain for the same 



TA S=TB S = 



TB C-T A G 

2 



, and 



SA G=TA G -{- 



TBG-TAG TAG+TBG 6 + e 



2 2 2 ' 

when we denote the tangential angles TA (7 and TB Ghyd and e, 
Kow since A C=BG=^AB~^ the chord s, we have 

^d + e\ 



F = ^ s" tang. (- 



2 



Fig. 42. 



G 



a/ 



This formula can be employed for the portion of surface 

M A B N, Fig. 42, whose tangential 
angles T A I) = a and T B F = fi 
are given ; putting the angle formed 
by the chord BAD = J^BE=zg, 
we have 

TAB ^ d = TA D ~- BAD 

z- a ~ a and 

TBA.^t^ABE-TBE 




>r 






— a -— 13, whence 
and the segment over A B 



F= I s' tang. {^^) 

or, since a — i3 is small, 

„ s' , ^, 6'* / tana, a — tang. (3 \ 

F= ^tang. («-/?) = j^ [ i +-fang. a tang.p l' 



84 INTRODUCTION TO THE CALCULUS. [Abt. 31. 

or since a and /3 differ but little from each other, and therefore we 

can substitute in tang, a tang. (3 instead of a and fi the mean value 

Oy we have 

,, , „ tang. a — tang. fi . ^ ^ ,. 

F = j\ s' . — i^qr^^rl^ = T2 « cos.' a (tang, a - tang, P), 

and substituting for s cos. a the base M N = x, 
F = j^ {tang, a — tang. (3), 

therefore the area of the entire portion of surface M A B N, when 
y^ and y^ designate its ordinates M A and N B, is 

^1 == (yo + 2/i) g + (^«^^. a - tang. (3) — . 

If another portion of the surface N B G adjoins the first and 
has a base N = x, and the ordinates B N and C — y^ and 
^2? and the tangential angles 8 B F = (3 and 8 G G = y, we have 
for the area of the same 

X x^ 

^2 = {yi +y^)^ + (tang. (3 - tang, y) — , 

and therefore for the whole surface, since— tang. (3 cancels + tang. /?,, 

F= F, + F2 = (iyo+yi + i y,) x + {tang, a - tang, y) ^. 

For a surface composed of strips of Hke width we have, when a 
is the tangential angle at the commencement and 6 at the end, 



x"^ 



■^= (i ^0 + 2/i + 2/2+ i 2/3) ^ + {tang, a - tang. 6) ^, 
and in general for a portion of surface, determined by the abscissas 

-, — ^, -^. . . X, and by the ordinates ^/o 2/i> ^2 • • • 2/«? ^^^ ^J ^^^ 
tangential angles Oq and a„ of the ends, 

^={iyo + yi+ y2 + •.. + yn-i+iyn)^ 

ill 

An Integral 

/ y dx = I (p{x)dx 

?= G yO + 2/1 + 2/2 + • • • + 2/n-l + i 2/«) - 

- I 



Art. 32.] 



INTRODUCTION TO THE CALCULUS. 



85 



can be found by putting x = Ci — c, calculating the values^ 

I ^ x\ ( nx\ , , . 

as well as tang, a = ^ = \j)(x)=il) (c) and tang. an= ijj (ci), and sub- 
stituting them in the equation. 

P^ d X 
E.G., for / — we have, if we take n = 6, since 

«/i X 

x — Ci — c—% —1 and 2/ =^^{x) =-^ 



also, since 



d y _d (x '^) _ 1 



d X 



d X 



X 



.2 y 



tang, a = — \ = —1 and ta7ig. /3= — (-)=— |, and therefore 



P'dx 
J \ X 



p4.6_L6t6i 6_t 6_|_1\ II. /_1 r 1\ _L _1 



4,1692 
G 



_ I . _i_ . _i_ = 0,69487 - 0,00173 = 0,69314. 



(Compare the example of the last article.) 

Akt. 33, To rectify a curve, or from its equation y r=^f[x>) be- 
tween the co-ordinates A M ::^ x and M P — y, Fig. 43, to deduce 
an equation between the arc A P = s and one or other of the 
co-ordinates, we determine the differential of the arc ^ P of the 
curve, and then we seek its integral. If x be increased by a quan- 
tity M]^=PP = dx, yis increased hjEQ = dy, and s by 

the element P Q = d s, and 
according to the Theorem 
of Pythagoras we have 



Fig. 43. 




P Q'^P P^^ QR\ 
I.E., 

d s^— d x^-\- d y", 

d s= Vdx^ + dy"^, 
hence the arc of the curre 
itself is 



M N 



=/" 



d x" -\- d ^^ 



86 INTRODUCriON TO THE CALCULUS. [Abt. 32. 

E.G., for Neil's parabola (see Art. 9, Fig. 17), whose equation is 
a'if^ t", we have % a y d y — 2> x^ d x, whence 

-m t} Jb (J/ Ju TTo U Jb CL Jb \j JO (a/ *J0 

dy — -^ and d y — 



"^ a y ^ 4:d' y"^ 4: a 

and d s"^ = ll -h - — ) d x\ hence 

\ 4: a} 



In order to find the necessary constant, we make s begin with x 
and y, and we obtain 

= .7^ a VX^ ■\- Con., or Con. = — -^^ a 



and s = rh a 



U,l 



[♦/(' * ^^■ - '} 



E.G., for the piece A Pi whose abscissa x — a,WQi have 



s =: -i^ a [ VJ^' - 1] = 1,736 a. 
Introducing the tangential angle QPR = PTM=a (Fig. 
43) vre have 

Q R = P Q.sin. Q P EandP R = P Q cos. Q P R, 
I.E., d y = d s sin. a and d x = d s cos. a, 

and besides, tang, a = --^ (see Art. 6), 

fj rji ft X 

also, sin. a — -j^ and cos. a = -^ ; and finally, 

d S CL s 

s = I Vl + tana fa . d x = / — -^ = / . 

If the equation between any two of the quantities x, y, s and a 
is given, we can find the equation between any two others. 

If, E.G., cos. a = — , we have 

Vc' + s' 

d X = d s COS. a = — , and 

Vc' H- s' 

/s d s . r % s d s , rd u , I , , . 

— =4 / — — ^ / —;-=\ / u~^du=u-^ 

Vc" + 5' ~^ i^c' + s' "^^ V«* ~*^ 

= Vc'' + 5^ + Const., and if a; and 5 are equal to zero at the 

same time, x = Vc^ + s"^ — c. 



Art. 33. 



INTRODUCTION TO THE CALCULUS. 



87 



Art. 33. A right line perpendicular to the tangent P T, Fig. 
44, is also normal to the curve at the point of tangency, for the 



Fig. 44. 




tangent gives the direction of the curve at this point. 

The portion P K oi the line between the point of tangency P 
and the axis of abscissas is called simply the Normal, and the pro- 
jection of the same M K on the axis of abscissas the Subnormal. 
We have for the latter, since the angle M P Kis equal to the tan- 
gential angle P T M — a, 

MK — MP . tang, a, 

I.E., the subnormal = y tang, a = y -A. 

tt Jb 

Since for the system of curves y = a;"", tang, a := m x"'~'^, it fol- 



lows that the subnormal is = m x" 



/y%Ttl 1 . /I'V) ^ "^ 



m y 

X 



and 



for the common parabola, whose equation is y"^ = p x, we have the 

subnormal = y ^ = -, that is constant. 
^ 2y 2 

If to a second point Q, infinitely near the point P, we draw 
another normal Q C, we obtain in the point of intersection of 
these two lines the centre C' of a circle which can be described 
through the points of tangency P and Q. It is called the circle of 
curvature, and the portions C P and C Q oi the normals are radii 
of this circle, or, as they are styled, the radii of curvature. This 
circle is the one of all those, which can be made to pass through P 
and Q, which keeps closest to the element P Q oi the curve, and 
we can therefore assume that its arc P Q coincides with the ele- 
ment P Q oi the curve. It is called the osculatory circle. 

Denoting the radius OP— (7 § by r, the arc ^ P of curve by 
s or its element P § by c? s, and the tangential angle or arc of 
P r if by a, and its element 8 U M-8 T M, le.,- US T=- 



88 



INTRODUCTION TO THE CALCULUS. 



[Akt. 33. 



P C 6 by ^ a, we have, since P Q=G P . arc of the angle P G Q, 

ds^—r da, whence the radius of curvatures /- = — —. 

da 

Fig. 45. 



U^ 




T A M N K ^ 

We can generally determine a from the equation of the co-ordi- 
nates by putting tang, a = -^. 

Now d tang, a = _ /^ , and cos. a = -=^, whence 



COS. a 



d s' 



d x^ 



da — cos? a . d tang, a = -=—^ d tang, a and 



ds' 






d s 



ds' 



cos.^ a d tang, a 



d s 



d x^ d tang, a' 
ds' 



Por a convex curve r = + -r^ = + ^ , -, , 

da d X d tang, a 

point of inflexion r — oo. 

Eor the co-ordinates A 
curvature, we have 



, and for a 



u and O = V of the centre C of 



Fig. 46 
T 



A, 



u=A M+H C=x-{- C P sin. G P H, i.e., u=x-i-r sin. a, and 
v=0 G=M P-H P=y-G P cos.G P H, i.e., v=y-r cos. a. 

The continuous line formed 
by the centres of curvature forms 
a curve, which is called evolute 
of A P, and whose course is de- 
termined by the co-ordinates u. 
and V. 

If the ellipse ADA, Dy, Eig. 
46, is laid upon the circle A B 
Ai Pi, its co-ordinates G M = x 
and M Q = g can be expressed 
by means of the central angle 
P G B — (pof the circle. We have 
here 



^^ 








/^s 


"^ 


^x. 


r 


^ 

y 


< 


V. 


\ c 




E-M. )A T ^ 


n\^ 


^ 


J 


^^1 


' 


.^ 




L 







Art. 33.] INTRODUCTION TO THE CALCULUS. 89 

X = C P sin. C P M — G P sin. B G P = a sin. 0, and 
y = MQ = - MP = ~ G P COS. G P M = i cos. 0. 

From the latter we obtain d x = a cos. (f) d (() and d y = — h 
sin. d (p, and consequently for the tangential angle of the ellipse 
Q TX= a 

. d y h sin. l , ,-,«., 

tana, a = -^ == — = ^«wfl'. 0, and fonts com- 

^ d X a COS. (p a ^ 

plementary angle Q T G = a^ = 180°— a, 

tang, a, = - tang. and cotg. ai = t cotg. 0.^ 

Hence the subtangent of the ellipse is 
MT= MQ cotg. MTQ 

= y cotg. «i = ^ cotg. = y, cotg. 0, 

when y^ designates the ordinate M P of the circle. Since the tan- 
gent P I^ to the latter is perpendicular to the radius G P, we have 
also P T M—P G B—cp, and therefore the subtangent M T of the 
same is also = MP cotg. MT P=yi cotg. 0. 

Therefore the two points of the ellipse and circle which have 
the same ordinate, have one and the same subtangent. 

Farther, for an elementary arc of the ellipse 

d s'=^ x^^dy""— {a" cos.^ (p +b'^ sin.^ 0) d (j)% 
and the differential of tang, a, 

d tang, a = d tang. = 



a a cos.^ 0' 

whence it follows that the radius of curvature of the ellipse is 

d s' _ (a' COS.' + ¥ sin.' 0)t 



r ~ 

(1 X' fl. tana, a , , 

a COS. . 



d x" d tang, a ^, ^^^^ 



a cos.^cp 

_ (a' cos.' (f) ■{- ¥ sin.' 0)1 
~~ ab 

E.G., for y = 0f I.E., for sin. = 0, and cos. = 1, we have the 
maximum radius of curvature 

__ a^ _ a"" 



90 INTRODUCTION TO THE CALCULUS. [Art 34. 

and, on the contrary, for = 90°, i.e., for sin. = 1 and cos. 
— 0, the minimum radius of curvature 

ao a 

The first value of r corresponds to the point i), and the last to 
the point A, and both are determined by the portions of the axes 
G L and C K, which are cut off by the perpendiculars erected upon 
the chord A^D dX its ends ^i and D. 

Aet. ^4. Many functions, which occur in practice, are com- 
posed of the various functions which we have already studied, 
such as 

y — x"^, y — e", and y = sin. x, y — cos. cc, etc. ; 

and it is easy, with the assistance of the foregoing rules, to deter- 
mine their properties, such as the position of their tangents, their 
quadrature, their radius of curvature, etc., as well as to construct 
the curves, as is shown by the following examples : 

For the curve, whose equation isy = x'll — o) = ^^ — I x^, 

we have d y = 2 x d x — x'^ d x, 

whence tang, a = 2 x ~ x"^ — x {2 — x). 

Since this tangent becomes ~ for a; = and 2: = 2, its direction 
at these two points is parallel to that of the axis of abscissas. 

Farther, d tang, a = 2 d x — 2 x d x = 2 {1 — x) d x, 
whence for x = 0, d tang, a = -\- 2 d x, 

and for x = 2, d tang, a = — 2 d x, 

and therefore the ordinate of the first point is a minimum, and that 
of the second point a maximum. If we put d tang, a = 0, we ob- 
tain X = 1 and y = ?^, the co-ordinates of a point of inflexion in 
which the concave portion of the curve joins the convex. 

Farther, for an element d s of the curve we have 
d s^ — d X' 4- d y' :=z dx" + x^ (2'— xy d x" — [1 + x"" (2 —xy] d x\ 
whence the radius of curvature is 

ds"" _ [1 + ^'^ (2 - xyY ^ 



r = 



d x' d tang a 2 (1 — 2;) 



i;.G., for X — we have r = -— = — ^, for x = 1, r = — -^ — 00, 
for cc = 2, r = -^ = + ^, and for x=3, r = i. 101= + 7,906. 



Art. 34.] 



INTRODUCTION TO THE CALCULUS. 



91 



The corresponding curve is shown in Fig. 47, in which A is the 

origin and XX, Y Y the 
axes of co-ordinates. The 
parabola B A B^, which ex- 
tends symmetrically upon 
both sides of the axis of A Yy 
represents the first v:\vt Pi=x'^ 
of the equation, an J, on the 
contrary, the curve C A (7i, 
which upon the riglit-hand 

side of Y JT descends below 

XX, and on the left-hand 
side rises above it, and thus 
diverges more and more from 

the axis X X, as it increases 

its distance from Y Y, cor- 
responds to the second part 




y^2 






In order to find for a given 
abscissa x, the corresponding 
point of the curve y = x' — 
^ x\ we have but to add alge- 
braically the corresponding 
ordinates of the fii'ct two 
curves ; e.g., since for x ~ 1 
we have ^i = 1 and ^.^ = — ^, 
it follows that the correspond- 
ing ordinate of the point W 
iS2j = i/, + y, = l -1=^; 

farther, for 2; — 2 we have 
^1 = 4, and y^ = — |> and 
hence the co-ordinate of the 
point M is y = 4: — § — ^. 
In the same way x = 3 gives 
2/ = yi +^2 = 9-9=::0;a:— 
— 1, t/= 1 -^ I = -j; x = -2, y = 4: + | = \^, etc., and we per- 
ceive that the curve from A towards the right has the form A W 
M K L, and that in the beginning it runs above the abscissa A K 
= 3, but from that point it extends to infinity below the axis 



_Y 



92 INTRODUCTION TO THE CALCULUS. [Art. 34 

X X, and that from A towards the left it forms but one branch 
A P Q . . ., which rises to infinity. From what precedes we see 
that TT is a point of inflexion, and M a point of the curve where 
the ordinate is a maximum. While the curve has in A and M the 

direction of X X,m W it rises at an angle of 45°, for we have for 
the latter tang, a = x {^ — x) = 1\ on the contrary, the angle of 
inclination at K, is tang, a = — 3, consequently a is = 71° 34', 
etc. The quadrature of the curve is given by the integral 

i^= J y dx = I {x^— ^x^) dx = jx^dx — \Jx^dx 

_ X^ X* _ x^ / x\ 

Hence, e.g., we have for the area of the portion of surface 
A WM XsiboYQ A K=d 



and on the contrary the area of the portion of surface 3 L 4 below 
the abscissa 3 4 is 

-P'. = y(l-|)-f (1-|) = 0-| = -|. 

Finally, to find the length of a portion of the curve, E.G., A WM, 
we put 

s— I Vl -{- x" {%— xY dx— I \(x)dx, 

and employ the method of integration explained in Art. 30, Here 

c is = 0, and d = 2, and taking ^ = 4 we have d x = — 

= — j— = ^, then substituting successively the values 0, i, 1, | and 



2 for X in the function ^ {x) = V 1 -\- x" {^ — xf, we obtain the 
values 

(0)=: 1^=1, <t> Q)= VU^^=^iA (1)= '^^1 + 1= 1^=1,414.... 

(/> (|)= 4/1+76=1 and (2)= Vl=l, 

and therefore the length of the arc A PT Jf is 

s = (0 (0)+4 a)+a <t> (l)+4 <!> (I) +0(2)) 1=1 

— (1+5+2,828 + 5 + 1). J = 3,471. 



Art. 35.] INTRODUCTION TO THE CALCULUS. 93 

By means of the curve y=x^ ll—A we can easily determine the 

course of the curve y=x y l—^hj extracting the square roots of 

o 

the values of the co-ordinates of the first, which give the corre- 
sponding co-ordinates of the latter. But since the square root 
of negative quantities are imaginary, this curve does not continue 
beyond the point K to the right ; and since every square root of a 
positive number gives two values, equal and with opposite signs, 
the new curve (//) runs in two symmetrical branches Q A M K 
and Qi A Ml K on both sides of the axis of abscissas. 

Akt. 35. When the quotient y = . ^ of two functions {x) 
and i/j (x) takes the indeterminate form of - for a certain value a 

Q^ Q^ 

of X, which always occurs when, as e.g., my = , the numer^ 

X Q/ 

ator and denominator of a fraction have a common factor ^ — «, 
we can find the real value of the same by differentiating the nu- 
merator and denominator. 

If X is increased by d x, and y by the corresponding element 
d y, we have 

, , <P {x) + d (j) {x) 

y -\- d y = , ; [ -,,;; , but tor x = a 

^ ^ ip (x) + dtp {xY 

(j) (x) = and V^ (x) = 0, whence 



, d (b (x) 
y + d y = -,.){ ; 
^ ^ dip (x)^ 



but since d y is infinitely small in comparison to y, we have 

_ (p (x) _ d(p {x) _ 01 (x) 
^ xp{x) d^(x) ip,{xy 

in which 0i (x) and ipi (x) designate the differential quotients of 

(x) and 'ip (x). 

(x) 

If ^ = -7^-^-, is also = -, we can differentiate it anew, and put 

_ d 01 (x) _ 02 (x) 
^ dip,{x) ip.ix)' 

In the same way the indeterminate expressions y = -^ and 



94 INTRODUCTION TO THE CALCULUS. [Art. 35. 

X 00, etc., can be treated, for oo = -, whence -^ and x oo 
can be put = - : 

Sx' - 11 x' - Sx -h 20 ^ 

E.G., y = - — I ~ — 2 . t T7 7 becomes iot x = 2, y = -. 

^ 5 x^ — 21 x^ + 24:X — 4: ' ^ 

For this we can put 

- ^^ (3 :^' - y ^' - 8 2; + 20) _ 9x' -Ux-S 
y ~ d (5 x'-21x' + 24:X-4:)~ 15 ^=^ - 42 a; 4- 24' 

which for 2; = 2 gives again y ^ -, and we can again put 

- ^^ (9x' -Ux - 8) _ 18 a? - 14 _ 9 a; - 7 _ 11 
^ ~ ^ (15 :i^' - 42 :« 4- 24) "" 30 :c - 42 ~ 15 ^ - 21 ~ 9* 

The factor (x — 2) is really contained twice in the numerator, 
and twice in the denominator. If we divide both by a; — 2, we 
obtain 

_ d x' — a; — 10 

^ ~ 5 o;^ - 11 ^ + 2' 

and dividing the last again by (x — 2) 

_ 3 X + 6 

y - sv^T 

which for :^ = 2 gives y = —, 



We have also for y = — when x = 0,-^, 

X \J 

^ d X 



but since d {a — Va^ — x) = — d {a' — x)^ — J 1 ~^ 

m this case y — , = ^r- : 

Va' - X ^a, 

I X . 

further y = , for x = 1, gives y — -x., 

Vl — X 

d X — d X 

but d I X = ~~ and d \^\ — x 



^ — -^ - 2 |/1 - a;' 



hence it follows that «/ = = ~- = 0. 

^ x 1 



Akt. 36.] INTRODUCTION TO THE CALCULUS. g$ 

-n- 11 1 — sin. X + COS. X . ^ n 

Finally, y = — ; -. ffives for a; = - (90°) 

-^ ^ — 1 + S171. X + COS. X ° 2 ^ ' 

y = — z — TT = 77, we have therefore 

^ — 1+1 + 0' 

d (1 — sin. X + COS. x) — cos. x — sin. x 

1] z=z ^^ zz: 

^ d{— 1 + sm X + COS. x) cos. x — sin. x 

0-1 "" ■^• 

Akt. 36. When, for a function y = au + l3v, a series of 
corresponding values of the variables ti, v and y has been deter- 
mined by observation or measurement, we can require the values 
of the constants a and (3 which are the freest from accidental or 
irregular errors of observation and measurement, and which 
express most exactly the relation between the quantities u, v and 
y, of which u and v are known functions of one and the same 
variable, x. Of all the metl-^ds that can be employed for the 
resolution of this problem, i.e., for the determination of the most 
possible, or the most probably correct, values of the constants, the 
method of the least squares is the most general, and rests upon the 
most scientific basis. 

If the results of the observations corresponding to the func- 
tion y = a u + (i V are, 

^1, ^1, yi 

^3j f^3f yz 



'^n) ^n} yn 

we have the following values for the errors of observation, and for 
their corresponding squares. 

Zi = yx — {au, + ^ v^) 
Zi = yc. — (a Ui -\- 13 v^) 
z% — yz — (a Ws + /3 v^ 



Zn — yn~ {a Un + ^ V^) 



96 INTRODUCTION TO THE CALCULUS. [Art. 36. 



zi'=yi^—2 a ill yi—2 13 Vi yi-{- a" Ui" + 2 a (3 Ui Vi + [3'' Vi 
Zi=y^— 2 au2 3/2—^ i^ ^^2 ^2 + ^^ u^ -^-2 a ^ u^_v<i-\- (3'^ Vo 
I %i=yi — 2 a Us ys—2 [3 Vz yz-^a" u^ -{-2a[3usVs + (3^ v, 



Zn=yn-2au^2/n—2fiv,,yr, + a'tC-{-2afiUnVn+f3''vJ' 

Employing the sign of summation 2 to denote the sum of 
quantities of the same kind, ^1" + y^^ + y^ + . . . -\- y^ = ^ (y"")^ 
Vi yi + v^ y^ + v^ y^i + . . . + v^ y,, — '^ {v y), etc., we have for the 
sura of the squares of the errors 

2 {z") = 2 (/) -2a'I.(uy) -2ii^{vy) + a"!, (u') 
+ 2al3l.(uv) + 13' 1 {v'). 

In this equation, besides the sum of the squares of the errors 
S (z^), which is to be considered as the dependent variable, only 
a and (3 are unknown. The method of the smallest squares 
requires us to choose such values for a and (3 as shall cause 2 (2;') 
to be a minimum ; and therefore we must differentiate the 
function 2 (2;'), which we have obtained, once in reference to a 
and once in reference to (3, and put each differential quotient 
of 2 («') thus obtained by itself equal to zero. In this way we 
obtain the following equations of condition for a and ft 

- 1 (uy) + a^ (u') + ^1 (uv) = 0, 

- 2 {v y) + (3 1, {v') + al.{uv) = 0, 

and resolving these we have 

^ 2 {v^) 2 in y) - 2 (u v ) S (vy) 

2 {ii') S {v') - 2 (^^ v) 2 \u v)' 

_ 2 {u') 2 {v y) -liuv)l {u y) ,a t 

/3 = ) / \-4^ — —) — f v-H\. See Ingemeur, page 77. 

2 (?r) 1 (v) — 1 [UV) 1 {UV) ° ' Jr o 

These formulas give for a function y — a + (3 v, since here 
w c,- 1, and 1 {uv) = 1 {v), 2 {u y) = ^ (y), and 2 {u") = 1+1 
+ 1 -^ ...z=z n, I.E., the number of equations or observations, 

n 1 {v') - 2 (^^) 2 (v) ' 
= n^ivy)-:Eiv):E(y) 
^ 7^ 2 (v') -l(v)l (v) ' 
Per the still simpler function y = 13 v,m which a = 0, we have 

2 (v') ' 



Art. 36.] 



INTRODUCTION TO THE CALCULUS. 



97 



and, finally, for the most simple case y — d, where we have to de- 
termine the most probable value of a single quantity, 

n 

that is the arithmetical mean of all the values found by measure- 
ment or by observation. 

Example. — In order to discover the law of a uniformly accelerated mo- 
tion, I.E., the initial velocity c and the acceleration p, we have measured 
the different times t^^ t^, ^3, etc., and the corresponding spaces «^, Sg, a^, 
etc., described, and have found the following results. 



Times . . . 





1 

3 

1 


5 


7 


10 sec. 


Spaces . . . 





5 


20 


38 


S8J 


10 1 feet. 



'Now ii 8 = ct ■] — ~ is the fundamental law of this motion, we are re- 
quired to determine the constants c and p. Putting in the foregoing for- 
mulas u =^ % and v =t^, and also a = c, [S = ~ and y = s, we obtain for 
the calculation of c and p the following formulas : 

2 (f) 2 (i*) - 2 {f) S {t^) 
p _ 2 {f) S (s f) — 2 {t^) 2 {8 t) 
2 ~ 2 (f) 2 (t') — 2 (f) 2 {t^) ' 

from which the following calculations can be made, 



t 


f 


f 


f 


s 


5i^ 


sf 


I 


I 


I 


I 


5 


5 


5 


3 


9 


27 


81 


20 


60 


180 


5 


25 


125 


625 


^38 


190 


950 


7 


49 


343 


2401 


58.S 


409.5 


2866.5 


10 


100 


1000 


1 0000 


lOI 


lOIO 


lOIOO 


Sum 


184 


1496 


13108 


222.5 


1674.5 


I4IOI.5 




= 1 (f ) 


-2(f) 


= 2(0 


= 2(s) 


=i:(st) 


= 2(5f). 



98 



INTRODUCTION TO THE CALCULUS. 



[Art. 37. 



from whicli we obtain 

- 1B108 . 1674,5 - 1496 . 14101,5 __ 85340 _ . QOft f + 
^ 184 . 13108 — 1496 . 1496 ~ 17386 ~ ' ' ^^^ 

_18 4 . 14101,5 - 1496 . 1674,5 _ ^962^ _ ^ 5155 feet 
^^ 184 . 13108 - 1496 . 1496 "~ 173860 ~ ^^^^^^ i^^^- 

Whence the formula for the observed movement is 

s = 4,908 t + 0,5155 f, 
and from this formula we have 



For the times . 





I 


3 


5 


7 


10 sec. 


For the spaces 





5-43 


19.36 


37-43 


59.62 


100.63 f^^t. 




Fig. 48, b If we consider the times 

(t) as abscissas, and lay off 
the calculated as well as 
the observed spaces (s) as 
ordinates, we can draw a 
curve through the extrem- 
ities of the calculated o'rdi- 
nates, which will pass be- 
tween the points Jf, iV, 0, P, 
Q, determined by the ob- 
served co-ordinates, so that 
the sum of the squares of the 
deviation of the curve from 
these points shall be as small 
as possible. 
Art. 37. If we have no formula for the successive values of a 

quantity y, or for its dependence 
upon another quantity x, and we 
wish to determine its value for a 
given value of cc, determined by 
experiment, or taken from a table, 
we employ the so-called method 
of interpolation, of w^hich only 
the most important part will be 
given here. 

If the abscissas A Mq = x^, 
A Mi = Xi and A M^ = x.^. Fig. 
49, and the corresponding ordi- 
nates Jfo A = 2^0, Mi Pi = 2/1, 
Jfg P2 = y^ are given, we can 



Fig. 49. 




Art. 37.] INTRODUCTION TO THE CALCULUS. 99 

express the ordinate MP=y, corresponding to the new abscissa A M 
—X, by the formula ?/=a +j3 2; -fy a;', provided three given points Po, 
Pi, P2, He nearly in a straight line or in a slightly curved arc. If we 
change the origin of co-ordinates from A to ifo? the generality of 
the expression will not be affected, and we obtain for x = ^ simply 
y — a, and consequently the constant member a = y^. Substi- 
tuting in the supposed equation, in the first place x^ and y^, and 
then in the second place x^ and y^, we obtain the two following 
equations of condition, 

y^ — y^ — {i Xx -\- y ^l^ and 
^2 — 2/0 = i3 ^2 + T ^2% hence 

Q ^ (^1 - y,) ^" - {y^ - yo) ^" ^^^ 
^ (^1 - yo) ^2 - (y2 - yo) x, 

' rp ^ rp _ ,, rp * rp 

from which we have 



rp^ 



V %7 ^ \ rp rp '^ ___ rp rp •* / \ rp ^ rp ___ /y» * /y* / 

If the ordinate yi lies midway between ^0 and y^, we have 0:3 = 
H Xiy and therefore more simply 



y-y^ \ %x, / "^ \ % X,' ) 



x^ 



If but two pair of co-ordinates x^, y^, and x^, y^ are given, we 
must regard the limiting line P^ Pi as a straight line, and conse- 
-quently put ^ = ^o + i^ ^ 

and y, ~ y^ + (i x^, 

whence we have ^ — ^ —, and 



*^i 



2/ = 2/0 + 



C^) - 



When it is required to interpolate by construction between 
three ordinates y^, y„ y.^ a fourth ordinate y, we draw, through the 
■extremities P^,, Pi, Po of these ordinates a circle, and take y —. to 
the ordinate of the same. The centre C of the circle is determined 
in the usual way by joining the points P^ Pi P2 by straight lines 
and erecting pei'pendiculars at the middle points of the chords. 
The point of intersection C of the perpendiculars is the required 
<;entre. 

If the distances of the middle point Pi from the two others Pq 



100 INTRODUCTION TO THE CALCULUS. [Art. 38. 

and Ps, are s^ and s^, and the distance Pi K of the point P from the 

chord Po F^ z=z Si = h, we have for the angle at the periphery 

a = Pi F^ Fi = ^ the angle at the centre Fi G F^ 

h 
sm, a = — , 

So 

and consequently the radius of curvature C F = C F^ — C Fi — 

^^-'^ ^ = ^a = \j' 

consequently we find the centre C of the circle passing through the 
points Po, Pi, Pg, by describing from P^ or Pi or F^ with a radius 
equal to the value of r, calculated by means of this formula, an arc 
whose intersection with the perpendicular to the chord F^ Pg erected 
at its centre D is the required point. 

*Akt. 38. The mean of all the ordinates upon the line M^ M^ is 
the altitude of a rectangle M^ M^ JV^ W^ with the same base M^ M^, 
and having the same area as the surface M^ Mo F^ Pi P^, and can 
therefore easily be determined from this surface. According to 
Art. 29 we have 

F — I y dx— I (y^ -^ (3 X + y x'^)dx 

ty ty 



ty 



= % ^2 + ^ + -^ 



— yO '^i -r \ ^ ^2 _ 2 / o 

f iyi - y.) ^2 - (y, - y,) ^A ^2' 



= (11 + fa ~ ^o) ^^' _ fa-yo) (3 a:i-2a:,) \ ^ . 
V° 6 ^1 (0:2 — ^1) 6 (2:2 — :^i) / ' 

_ (yo + ^2\ , / fa - yo) ^2 - (.^2 - yo) xA , 
-\ 2 7^^"^ V 6^1(^2-^1) ) '' 

and consequently the mean ordinate is 

Z ^ iMo_±jA + / fa-^o)^^-fa-yo)^A ^^. 

3/2 /C \ O iCi (3/2 '^1/ / 

If ^ ^ were = — , the boundary would be a right line, and 

we would have simply 

and ^m - ^^" ^ ^' . 



Art. 38.] 



INTRODUCTION TO THE CALCULUS. 



101 



Fig. 50. 



If also Xi = 2 Xi, that is, if i/i is equidistant between y^ and 
^2, we have 

i^ = (^0 + 4 yi + y^) ^ (see Art. 30), and i/,, =y±J^—lL±ll, 

If a surface M, M, P, P„, Fig. 
50, is determined by four co-or- 
dinates M, P, = y„ Jfi Pi = y,, 
Mc, Po = y^, Mi P3 =: ?/3, which 
are equidistant from one an- 
other, we can determine approx- 
imately the area of the same in 
the following simple manner : 

Let us denote by 0:3 the base 
Mq M3, by z^ Zi Zo,, three ordinates 
intercalated between y^ and y^, 
and equidistant from each other. 







N1M.1 N^ MaNs 
we can then put approximatively the surface 



2/Q 



M, M,P,P, = F= (i y^ + z,+z,+ z,+ }, y,) -^; but 

Zx + z<i -\- Zz 2zi + 2z.2-h2zs 2 Zi + z^ 2 Z3 + z^ , 
~ ~ -\ and 



6 



6 



6 



, 1 / . 2zx + z^ 2 Zs -{• z, 
yi = Zi + i (^2 - Zx) = , as well as y, = ^ — 



whence it follows that 



Z\ -{• Z<i -J- z^ 



^^ and 



X^ 



F^[\y. + i{y, + y.) + hy.]^ 



= [i/o + 3 (2/1 + y^ + y^ y, and also 
,, _ yo + 3 {y, + y,) + ^3 

yra- g 

While the former formula for ?/„ is employed when the surface 
is divided into an even number of strips, the latter is employed 
when the number of these divisions is uneven. 

Hence we can write approximately 



/ y dxr^ / (.t) f?a; = [2/„ + 3 («/, H- y.^ + y^ 



Cl 



8 






102 INTRODUCTION TO THE CALCULUS. [Art. 88. 

%=^1> (ohVx = (f> ( — 3— j, 2/2 = (^ 3 j and 2/3 = <^ (ci) are 

— (see example, Art. 

30) we have c = l,Ci = 2 and (x) = -, whence it follows that 

3 3 

the approximate value of this integral is 



PAET FIEST. 



GENERAL PRINCIPLES OF MECHANICS. 



FIRST SECTION. 

PHORONOMICS OR THE PURELY MATHEMATICAL 

THEORY OF MOTION. 



CHAPTER I. 

SIMPLE MOTION. 

Y-1 



§ 1. Rest and Motion. — Everybody occupies a certain posi- 
tion in space, and a body is said to be at rest, (Fr. repos, G-er. Ruhe), 
when it does not change that position, and, on the contrary, a body 
is said to be in motion, (Fr. mouvement, Ger. Bewegung), when it 
passes continually from one position to another. 

The rest and motion of a body are either absolute or relative, 
according as its position is referred to a point which is itself at rest 
or in motion. 

On the earth there is no rest, for all bodies upon it participate 
in its motion about its axis and around the sun. If we suppose 
the earth at rest, all the terrestrial bodies which do not change 
their position in regard to the earth are at rest. 

§ 2 Kinds of Motion. — The uninterrupted succession of po- 
sitions which a body occupies in its motion forms a space, that is 
called the path or trajectory (Fr. Chemin, trajectoire, Ger. Weg) of 
the moving body. The path of a point is a line. The path of a 
geometrical body is, it is true, a figure, but we generally under- 
stand by it the path of a certain point of the moving body, as, e.g., 
its centre. Motion is rectilinear (Fr. rectiligne, Ger. geradlinig) 



106 GENERAL PRINCIPLES OF MECHANICS. [§ D-5. 

when the path is a right line, and curvilinear (Fr. curviligne, Ger. 
ki'ummlinig) when the path of the moving hody is a curved hne. 

§ 3. In reference to time (Fr. temps, Ger. Zeit) motion is either 
uniform or varialle. Motion is uniform (Fr. uniforme, G. gleich- 
formig) when equal spaces are passed through in equal arbitrary 
portions of time. It is variable (Fr. varie, Ger. ungleichformig) 
when this equality does not exist. When the spaces described in 
equal times become greater and greater as the time during which 
the body is in motion increases, the variable motion is said to be 
accelerated (Fr. accelere, Ger. beschleunigt) ; but if they decrease 
more and more with the increase of time, this motion is said to be 
retarded (Fr. retarde, Ger. verzogert). Periodic (Fr. periodique, Ger. 
periodisch) motion differs from uniform motion in this, that equal 
spaces are described only within certain finito spaces of time, which 
are called periods. The best example of uniform motion is given 
by the apparent revolution of the lixed stars, or by the motion of 
the hands of a clock. Examples of variable motion are furnished 
by falling bodies, by bodies thrown upwards, by the sinking of the 
surface of water in a vessel which is emptying itself, etc. The 
play of the piston of a steam engine, and the oscillations of a pen- 
dulum, afford good examples of periodic motion, 

§ 4. Uniform Motion. — Velocity (Fr. vitesse, Ger. Geschwin- 
digkeit) is the rate or measure of a motion. The larger the space 
that a body passes through in a given time, the greater is its mo- 
tion or its velocity. In uniform motion the velocity is constant, 
and in variable motion it changes at each instant. The measure 
of the velocity at a given moment of time is the space that this 
body either really describes, or which it would describe, if at that 
instant the motion became uniform or the velocity remained con- 
stant. We generally call this measure simply the velocity. 

§ 5. If a body in each instant of time describes the space o, and 
if a second of time is made up of n (very many) such instants, 
then the space described within a second is the velocity, or rather 
the measure of the velocity, and it is 

c = n . (7. 

During a time t (seconds) n » t instants elapse, and in each in- 



^ 6, 7.] SIMPLE MOTION. 107 

stant the body passes through the space cr, and therefore the total 
space, (Fr. Tespace, Ger. Weg), which corresponds to the time t, is 

sz^n.t.o — n.o.t, I.E. 

I, ) s — c t. 

In uniform motion the space (s) is a product of the velocity (c) 
a7id the time (t). 

Inversely II.) c = -. 

III.) t ^ -. 

c 

Example. — 1. A locomotive advancing with a velocity of 30 feet passes 
in two hours = 120 minutes = 7200 seconds, over the space « = 30 . 7200 
= 216000 feet. 

2. If we require 4|- minutes = 270 seconds to raise a bucket out of a 

pit, which is 1200 feet deep, we have its mean velocity (c) = -^^ = — 

= 44 = 4,444 . . . feet. 

3. A horse advancing with a velocity of 6 feet requires, to pass over five 

miles, or 26400 feet, the time t = — ^ — = 4400 seconds, or 1 hour 13 

o 

minutes and 20 seconds. 

§ 6. If we compare two different uniform motions, wc obtain 
the following result : 

As the spaces are s ='C t and Si — Ci ti their ratio is - = — j. 

Si Cx ti 

s c s 

If we put t = tiwe have - = - ; if we take c = CiWe obtain - = 

Si C\ S\ 

t ' . c t 

- ; and finally, if s = Si it follows that ~ = -. 

t\ C\ t 

Tlie spaces described in the same time in different uniform mo- 
tions are to each other as the velocities ; the spaces described with 
equal velocities are to each other as the times ; and the velocities cor- 
responding to equal spaces are inversely as the times. 

§ 7. Uniformly Variable Motion. — A motion is uniformly 
variable, (Fr. uniformement varie, Ger. gleichformig verandert), 
when the increase or diminution of the velocity within equal, ar- 
bitrarily small, portions of time is always the same. It is either 
uniformly accelerated (Fr. uniformement accelere, Ger. gleichfor- 



108 GENERAL PRINCIPLES OF MECHANICS. [§ 8, 9, 

mig beschleunigt) or uniformly retarded (Fr. uniformement retarde, 
Ger. gleichformig verzogert). In the first case a gradual augmen- 
tation, and in the second a gradual diminution of velocity takes 
place. 

A body falling in vacuo is uniformly accelerated, and a body 
projected vertically upwards would be uniformly retarded, if the 
air exerted no influence upon it. 

§ 8. The amount of the change in the velocity of a body is 
called the acceleration (Fr. acceleration, Ger. Beschleunigung and 
Acceleration). It is either positive (acceleration) or negative (re- 
tardation), the former when there is an increase, and the latter 
when there is a diminution of velocity. In uniformly variable mo- 
tion the acceleration is constant. We can therefore measure it 
by the increase or decrease of velocity which takes place in a 
second. For any other motion, the acceleration is the increase or 
decrease of velocity, which a body would undergo if, from the instant 
for which we wish to give the acceleration, the acceleration became 
constant, and the motion was changed to a uniformly varied one. 

This measure is generally called simply the acceleration. 

§ 9. If the velocity of an uniformly accelerated motion in a very 
small (infinitely small) instant of time is increased by a quantity 
K, and if the second of time is composed of n (an infinite number 
of ) such instants, the increase of velocity in a second, or the so- 
called acceleration, is 

p =z n K, ' 

and the increase after t seconds i^ = nt,ii=^nK,.t=pt. 

If the initial velocity (at the moment from which we begin to 
count t) is = c, we have for ^q final velocity, I.E., for the velocity 
at the end of the time t, 

V = c + pt. 

For a motion starting from rest c is = 0, whence v — pt ; and 
when the motion is uniformly retarded, in which case the accelera- 
tion ( — i? ) is negative, we have 

V = c — p t 

Example. — 1. The acceleration of a body falling freely in vacuo is 
= 32,30 feet. It acquires therefore after 3 seconds the velocity v = pt =z 
32,20 . 3 = 96,60 feet. 

2. A ball rolling down an inclined plane has in the beginning a velocity 



§ 10.] SIMPLE MOTION. 109 

of 25 feet, and the acceleration is 5 feet per second. Its velocity after 2|- sec- 
onds is therefore « = 25 + 5 . 3,5 = 37,5 feet ; i.e., if from the last moment 
it moved forward uniformly, it would pass over 37,5 feet in every second. 

3. A locomotive moving with a velocity of 30 feet loses, in consequence 
of the action of the brake, 3,5 feet of its velocity every second ; its accelera- 
tion is therefore — 3,5 feet and its velocity after 6 seconds is » = 30 — 3,5 . 6 
= 30 - 21 = 9 feet. 

§ 10. Uniformly Accelerated Motion. — Within an infinitely 
small instant of time r we can consider the velocity of every 
motion as constant, and put the space passed through in this 
instant 



a z=z V . r. 



f 



and we obtain the space passed through in the finite time t by 
summing these small spaces. But the time in which all these 
small spaces were described is one and the same r, and we can put 
their sum equal to the product of this instant of time and the sum 
of the velocities corresponding to the different equal instants. 

For uniformly accelerated motion the sum (0 + v) of the ve- 
locities in the first and last instant is just as great as the sum 
p T + {v — p t) of those in the second and last but one instants, 
and equal to the sum 2 p r + {v — 2p r) of those in the third and 
last but two instants, etc., and this sum is in general equal to v ; 

the sum of all these velocities is therefore equal to (^ • n) ^^^ Pro- 
duct of the final velocity and half the number of the elements 
of the time, and the space described is equal to the product 

Iv.^.r J of the final velocity v and half the number of the elements 

of the time and one of these elements. Now the magnitude (r) 

of an element of the time multiplied by their number gives the 

•whole time ^, whence the space described in the time f with an 

vt 
uniformly accelerated motion \s s = -^• 

The space described with uniformly accelerated motion is the 
same as that described with uniform motion when the velocity of 
the latter is half the final velocity of the former. 

Example. — 1. If a body in uniformly varied motion has acquired in 10 
seconds a velocity « = 26 feet, the space described in the same time is 

i = — - — = 130 feet. 
2 



110 GENERAL PRINCIPLES OP MECHANICS. [§11,13. 

3. A wagon whose motion is uniformly accelerated and wliich describes 

25 feet in 2^ seconds, possesses at the end of that time the velocity 

2.25 50.4 __ „ , 
V = = — ^— = 22,22 . . . feet. 

§ 11. The two fundamental formulas of uniformly accelerated 

motion 

I.) V =p t and 

^ vt 

U.)s = ^, 

which show that the velocity is a product of the acceleration and 
the time, and that the space is the product of half the terminal ve- 
locity and the time, furnish two other equations, when we eliminate 
in the first place v and in the second t. By this operation 
we obtain 

III.) s = IT- and 

IV.) s = ^. 

Hence, in uniformly accelerated motion, the space descriled is 
equal to the product of half the acceleration and the square of the 
time, and also to the square of the terminal velocity divided hy dou- 
Ue the acceleration. 

From these four principal formulas we deduce by inversion, 
and by the elimination of one or other of the quantities contained 
in them, eight other formulas, which are collected together in a table 
in the " Ingenieur," page 325. 

Example. — 1. A body moving with the acceleration 15,625 feet, describes 

15 625 (1 5)^ 9 

in 1,5 seconds the space — ^ ^ — —■ = 15,625 . — = 17,578 feet, 

til o 

2. A body, which acquires a velocity «) = 16,5 in consequence of an 
acceleration p = 4,5 feet, has described in so doing the space s = 

^4^ = 30,25 feet. 
2 . 4,5 

§ 12. On comparing two different uniformly accelerated mo- 
tions, we arrive at the following conclusions. 

The velocities are v — p t and v^ = P\ t^. The spaces, on 

7) f^ r) f "^ 

the contrary, are s = ~r- and Sx — —5^, whence we have 

• — — 7- and. • — 7-^ — 7- — 2 • 

Vi pi tx Si px tx Vi ti Vi p 

Putting ^1 = ^ we obtain : 



§13.] SIMPLE MOTION. HI 

S V v • 

— = — == — ; the times being equal, the ratio of the spaces de- 

Si Vi pi 

scribed is equal to that of the final velocities or of the accel- 
erations. 

If we put pi = p we have 

V t ^ s f v^ 
— — — and — = — = -— . 

Vx h Si ti Vi 

The acceleration being the same, i.e., when we have the same 
uniformly accelerated motion, the final velocities are to each other 
as the times, the spaces described as the squares of the times, and 
also as the squares of the final velocities. 

Farther, if we take ^^i = ?; it ffives — = -^ and — = -^: for the 

Pl t cSl tl 

same final velocities the accelerations are to each other inversely, 
and the spaces directly as the times. 

7) t "^ V^ 

Finally, for ^i = 5 we have -^ = -^ = -^ ; for equal spaces de- 

P\ t Vi 

scribed the accelerations are to each other inversely as the squares 
of the times and directly as the squares of the velocities. 

§ 13. For a uniformly accelerated motion with the initial veloc- 
ity c we have from § 9 

I.) V = c + p t, 

and since the space c t belongs to the constant velocity c, and the 
space ^-^r- to the acceleration p 

II.) S= Ct + ^ 



2 
Eliminating p from the two equations, we obtain 

c ■\- V 
III.) s = -^— t, 



v' -c' 



or ehminating t, we find 

IV.) s= ^ . 

2 p 

Example. — 1. A body moving with the initial velocity c = 3 feet and 
with the acceleration p = 5 feet describes in 7 seconds the space 

8=3.7 + 5.-^ = 31 + 122,5 = 143,5 feet. 

2. Another body, which in 3 minutes = 180 seconds changes its ve- 
locity from 2^ feet to 1^ feet, describes during this time the space 

?l^-4^. 180 = 900 feet. 



112 GENERAL PRINCIPLES OF MECHANICS. [§ 14. 

§ 14. Uniformly Retarded Motion. — For uniformly retarded 
motion with the initial velocity c we have the following formulas, 
which are deduced from those of the foregoing paragraph by mak- 
ing p negative. 

L) v = c—pt, 

111.). = ^^.^, 



IV.) s = 



& — -y' 



2j9 



While in uniformly accelerated motion the velocity increases 

without limit, in uniformly retarded motion the velocity decreases 

up to a certain time, when it is = 0, and afterwards it becomes 

negative, I.E., the motion continues in the opposite direction. 

If we put V = in the first formula, we obtain p t ~ Cy whence 

c ' 
the time in which the velocity becomes = is ^ = — ; 

substituting this value of t in the second equation, we obtain the 

space described by the body during this time, s = -^ — . 

c c^ 

If the time is greater than — , the space is smaller than ^— ; 

2 c 
and if the time is = — the space becomes — 0, the body having re- 

jj 

turned to its point of departure ; finally, if the time is greater than 

2 c 

— , s is negative, I.E., the body is on the opposite side of the point 

V 

of departure. 

Example. — A body which is rolled up an inclined plane with an in- 
itial velocity of 40 feet, and which suflfers a retardation of 8 feet per sec- 

40 . 40^^ 

ond, rises only during — = 5 seconds and reaches a height of - — - = 100 

8 2.8 

feet, after which it rolls back and arrives after 10 seconds with a velocity 
of 40 feet at the point from whence it started, and after 12 seconds is al- 
ready 40 . 12 — 4 . 12^^ or - (40 . 2 4- 4 . 2*) = 96 feet below its point of de- 
parture, if the plane continues beneath H. 



§15,16.] SIMPLE MOTION. 113 

§ 15. The Free Fall of Bodies. — Th^free or vertical fall of 
bodies in vacuo (Fr. mouvement vertical des corps pesants, Ger. 
der freie oder senkrechte Fall der Korper) furnishes the most im- 
portant example of uniformly accelerated motion. The acceleration 
of this motion produced by gravity (Fr. gravite^, Ger. Schwer- 
kraft) is designated by g, and its mean value is 

9,81 meters. 

30,20 Paris feet. 

32,20 English feet. 

31,03 Vienna feet. 

31^ = 31,25 Prussian feet. 

32,7 Bavarian or meter feet. 

If any of these values of g be substituted in the formulas v=g t, 

s = '^^ and s = ^, v = V2 g s, all possible questions in relation 

to the free fall of bodies can be answered. 

For the metrical system of measures we have 

V = 9,81 . t = 4,429 Vs, 
s = 4,905 f = 0,0510 v\ 

t ^ 0,1019 v' = 0,4515 Vs; 

and for English measures 

V = 32,2 t = 8,025 Vs, 
s = 16,1 f = 0,0155 v\ 

t = 0,031 v = 0,249 V^, 

Example. — 1.) A body attains when it falls unhindered in 4 seconds a 
velocity « = 33,3 . 4 = 138,8 feet, and describes in this time the space s = 
16,1 . 4^ = 357,6 feet. 3.) A body which has fallen from the height s = 

9 feet, has the velocity v = 8,035 4/9 = 34,075. 3.) A body projected ver- 
tically upwards with a velocity of 10 feet rises to the height s = 0,0155 . 
10^ = 1,55 feet, in the time 

t = 0,031 . 10 = 0,31, 

or nearly |^ of a second. 

§ 16. The following Table shows how the motion takes place as 
the time elapses, 



IM 



GENERAL PEINCIPLES OF MECHANICS. 



[§17. 



Time in) 
seconds j 


o 


I 


2 


3 


4 


5 


6 


7 


8 


9 


lO 


Velocity. 


o 


-^9 


^9 


Z9 


4i7 


5^ 


6^ 


75^ 


8^ 


99 


lO^ 


Space 


o 


2 


i 


^f 


16^ 

2 


^3f 


3« 


-f 


< 


8i^ 

2 


lOO- 
2 


! Difference 


o 


2 


A 


A 


'! 


A 


11^ 

2 


-f 


< 


-! 


-! 



The last horizontal column of this table gives the spaces de- 
scribed by a body falling freely in each single second. We see that 
these spaces are to each other as the uneven numbers 1, 3, 5, 7, etc., 
while the times and the velocities are to each other as the regular 
series of numbers 1, 2, 3, 4, 5, etc., and the distances fallen through 
as their squares 1, 4, 9, 16, etc. Whence, e.g., the velocity after 6 
seconds is = 6 ^ = 193,2 feet, i.e., the body, if from this moment 
it continued to move uniformly as on a horizontal plane which of- 
fered no resistance, would describe in every second the space 6 ^ = 
193,2 feet. It does not really describe this space in the following 
or seventh second, but from the last column we see that it de- 

n 

scribes exactly 13 ^ — 13 . 16,1 = 209,3 feet, and in the eighth 

second 15 f = 15 . 16,1 = 241,5 feet. 

Remark. — Older German writers designate the space 16,1 feet, de- 
scribed by a body falling freely in the first second, by ^, and call it also the 
acceleration of gravity. They employ for the free fall of bodies the for- 
mulas 



8 = gf = 



4^' 






g 



This usage, known only in Germany, is tending gradually to disappear, 
which, on account of the frequent misapprehensions and errors resulting 
from it, is much to be desired. 



§ 17. Free Pall with an Initial Velocity.— If the free fall 
of a body takes place with an initial velocity (Fr. vitesse initiale, Grer. 



1 17.] SIMPLE MOTION. 115 

Anfangsgeschwindigkeit) c, the formulas assume the following 
form ; 

v = c-{-gt = c + 32,2 t feet = c -h 9,81 t meters, 

V = Vc' + 2 g s = 4/?~+ 64,4^ feet = Vc' + 19,6"25 meters, 

s=:c^+ff = c^f + 16,1 f feet = c 2^ + 4,905 f meters, 

2 2 

and s = ^ ~ ^ = 0,0155 (t;^ - c') feet = 0,0510 (v' - c') meters. 
2 g ^ ^ ^ ^ 

If, on the contrary, the body is projected vertically upwards, we 
have 

v = c— gt = c — 32,2 f feet — c — 9,81 t meters, 

V = V? -2 gs = Vc' - 64,4 5- feet = V r - 19,62 5 meters, 

5=zci^-|f=:cz^- 16,1 z^^ feet = ct - 4,905 f meters, 

2 2 

and 5 = --^^ = 0,0155 (c^ - v') feet = 0,0510 (c' - v') meters. 

2^ ' ^ \ / 

If we consider a given velocity c as a velocity acquired by a free 
fall, we call the space fallen through 

^ = 0,0155 c' feet = 0,0510 c' meters, 
2g 

"the height due to tlie velocity" (F. hauteur due a la vitesse, G-er. 
Geschwindigkeitshohe). By the substitution of the above, several 
of the foregoing formulas may be expressed more simply. If we 

denote the height \^r-) due to the initial velocity by h, and that 



© 



9^ 
due to the final velocity by h, we have for falling bodies, 



9 

h = /t; + 5 and s = h — h, 

and for ascending bodies, 

h = Jc — s and s = k — h. 

Tlie space described in falling or ascending is therefore equal 
to the difference of the heights due to the velocities. 

Example. — If for a uniformly varied motion the velocities are 5 feet 
and 11 feet, and the heights due to the velocities are 0,0155 . 5^ = 0,3875, 
and 0,0155 . ll'^ — 1,8755,. the space described in passing from one velocity 
to the other is s = 1,8755 — 0,3875 = 1,4880 feet. 



116 GENERAL PRINCIPLES OF MECHANICS. [§18. 

& — v^ 

§ 18. Vertical Ascension. — If in the formula s = —x, 

2g 

for the vertical ascension of bodies we put the final velocity v = 

0, we obtain the maximum height of ascension, 

_^ 

consequently the maximum height of ascension, corresponding to 
the velocity c, is equal to the height of fall Jc due to the final velo- 
city c, and therefore c = V2 g k is not only the final velocity for 
the height k of free fall, but also the ■ initial velocity for the maxi- 
mum height of ascension h Hence it follows that a body pro- 
jected vertically upwards has at any point the same velocity, which 
it would have, in the opposite direction, if it fell from a height 
equal to the remaining height of ascension to that point, and which 
it really possesses afterwards, when it reaches it upon falling back. 

ExAiAPLE. — A body projected vertically upwards, with a velocity of 15 
feet, after ascending 2 feet meets an elastic obstruction, wMch throws it 
back instantaneously with the same velocity with which it struck. How 
great is this velocity, and how much time does the body require to ascend, 
and fall back again ? The height due to the initial velocity 15 feet is A; =r 
3,49 feet, and the height due to the velocity at the instant of collision is 
h = 3,49 — 3,00 = 1,49, and, consequently, the velocity itself is = 8,025 

4/1,49=9,8 feet. The time necessary to ascend the entire height (3,49 feet) 
would het = 0,031 c = 0,031 . 15 = 0,465 seconds, while the time neces- 
sary to ascend the height 1,49 is ^j = 0,031 . 9,8 = 0,3038 seconds, whence 
|:he time necessary to ascend the 2 feet h t — t^ -~ 0,465 — 0,3038 = 
D,1612 seconds, and finally the whole time employed in ascending and fall- 

3224 
ing is = 2 . 0,1612 = 0,3224 seconds. This, therefore, is but qokt^, or 

about ^ of the time, which would be employed by the body in rising and 
falling if it met with no obstacle. This case occurs in practice in forging 
red-hot iron, for we are obliged to give as many strokes of the hammer 
as possible in a short space of time, on account of the gradual cooling of 
the iron. If by means of an elastic spring we cause the hammer to be 
thrown back, it can, under the circumstances supposed in the example, 
make three times as many blows as when its rise was unimpeded. 

Remakk 1. — In practical mechanics, particularly in hydraulics, we are 
often obliged to convert velocity into height due to velocity, or the latter 
into the former, A table, by means of which this operation can be per- 
formed at once, is of the greatest service to the practical man. Such a 
one. Calculated for the Prussian foot, is to be found in the " Ingenieur,"' 
page 326 to 329. 



§19.] SIMPLE MOTION. II7 

Remark 2. — The formulas deduced in the foregoing paragraphs are 
strictly correct only for bodies falling freely in vacuo; they are, however, 
sufficiently accurate for practical purposes, when the weight of the body is 
great compared to its volume, and when the velocities are not very great. 
They are, besides, employed in many other cases, as will be shown here- 

§ 19. Variable Motion in General. — The formula s = ct 
(§5) for uniform motion holds good also for every variahle motion, 
if instead of t we substitute an element or an infinitely small in- 
stant of the time r, and instead of s the space g described in this 
instant, for we can assume that during the instant r the velocity c, 
which we here denote by v, remains constant, and that the mo- 
tion itself is uniform. 

Hence, we have for every variable motion 

I.) G ^z V r, and V = - (compare § 10). 

T 

TJie velocity {v) for every instant is given by the quotient of the 
element of the space divided ty that of the time. 

In like manner the formula v = p t {^11) iov uniformly accele- 
rated motion holds good also for every variable motion, if instead of 
t and V we substitute the element of time r and the infinitely small 
increase of velocity n during that time, for the acceleration p 
does not vary sensibly in an instant r, and the motion can be re- 
garded as uniformly accelerated during this instant. Consequently 

we have for all motions 

If 
II.) K = p r, and p =^ -. 

The acceleration ( p) is, therefore, equal to the element of the ve- 
locity divided hy the element of the time. 

If we put the total duration of the motion t — nr^ and the ve- 
locities in the successive instants r are v^, v^, v^ . . v,„ the corres- 
ponding elements of the space are g^ — Vi r, g^ ^ v^ t, g^ = v-s r . . ^ 
^n — v„ r, and the total space described is 

s = {vi + v, + Vs. . .v„) T = ^^ 'j n T, I.E., 

P) s = i ^1 t = vt, when 

V = ^ — ^ '^3---^" denotes the mean velocity of the body while 

n 

d-scribing the space 5. 



118 GENERAL PRINCIPLES OF MECHANICS. [§ 19. 

In like manner if c denotes tlie initial and v tlie final velocity, 
and if jOij^g > ' > Pn denote the accelerations in the equal successive 
instants r, we have 

V — c — {pi + pi + .. .p„) T =z l^ ^ —^—j n T, I.E., 

IP) v-c= (^L±^_-^ t -_= p I ^hen*V 

__/^i — yj — ^._. — Pn ^gj^Q^gg ^\^Q mean acceleration. 

n 

By combining the formulas I. and II. we obtain the following 
not less important equation : 

III.) V K, -= ^ a. 

If, while the space s^= no '\^ described, the acceleration assumes 
successively the values px, p^ . . .p„, the sum of the products p o is 

(p,+p,...+p.)o = (^Pl±Pi±lL:±&y a 

If the initial velocity c is transformed by successive increases 

V — c 
of K — into the final velocity v, the sum of the products 

V K is 

C K -{- {C -r H^) K -^ + (V — tc) K -\- V K = [c + {C + ic) + . . .-[-{v — fc) + v] K 

, . 71 K (V -\-c) (V — C) V^ — & 

and therefore we can write 

-y^ Q^ qjl"^ (f 

IIP) — ^ — = ^ 5, or s = — ^ (compare IV., § 13). 

With the aid of the foregoing formulas we can solve the most 
varied problems of phoronomics and mechanics. 

The time, in which the space s = 7^ cr is described with the vari- 
able velocities Vx, i\, . . . v„, is 

TTTX^ /I 1 1\S/1 1 1\S 

\Vi v^ vj n \vi V.2 v„/ V 

when we put the value - I 1 h . . . H ) = -> whose recip- 

n \Vx V.2 v„ / V 

rocal V can be considered as the mean velocity. 

Example. — When a body moves according to the law v =at^, we have 

V -\- K = a {t + rf — a {f^ + 2 t T + r-), and k = a t (2 t + t), consequently 

p= - = 2 at. 



§30.] SIMPLE MOTION. 119 

The velocities of the body at the end of the times 

T, 2t^ d T . ..n T are a r^, a (2 ry, a {^ ry . .a in rf, 
whence it follows that the space described \iQ.t^=7iT seconds is 

« = [« r^ + « (2 r)2 + . . a (7i ry] r = (1^ + 2" + 3'^ + . . + rh") a t\ 
but from Article 15, IV., of the Introduction to the Calculus we have 

12 + 2^ + 3^ + . . + 71^ =-- ^, hence 

o 

* = ^^" = 3^^^^)^-T 

(§ 20.) Differential and Integral Formulas of Phoronc- 
mics. — The general formulas of motion found in the foregoing 
paragraphs assume, when the notations of the calculus are em- 
ployed, I.E., when the element of time r is designated by d t, the 
element of space ohj d s, and the element of velocity k, by d v, the 
following form : 

I.) V = -^j—,, or d s = V d t, whence s = I vdt, and t = / — . 
d t ^ ^ V 

II.) p = —- OY d V =pd t, whence v — I pdt, and t= I —-. 

III.) V d V = p d s, or s = / — — , and — - — = I p d s, 

in which c denotes the initial and v the final velocity, while the 
space s is being described. 

We see from the above that the difference of the squares of the 
velocities is equal to tioice the integral of the pjroduct of the accelera- 
tion and the differential d s, or equal to the jiroduct of the mean ac- 
celeration and the space described ty the hody in passing from the 
velocity c to the velocity v. 

According to the theory of maxima and minima the space is a 

maximum, and the motion attains the greatest extension, when we 

have 

d s ^ 

J7 = " = •'' 

and the velocity is a maximum or minimum when 

dv 

rt = ^ = ^- 

The foregoing are the fundamental formulas of the higher 
Phoronomics and Mechanics. 

Example, — 1. From the equation for the space s = 2 + 3 i + ^^ we 
dedu:;e by differentiation the equation for the velocity «>=3 + 2 ^, and that 



120 GENERAL PRINCIPLES OF MECHANICS. [§20. 

for the acceleration p = 2; the latter is constant and the motion is uni- 
formly accelerated, 

For i = 0, 1, 2, 3 . . . seconds, we have 

« = 3, 5, 7, 9 . . . (Feet), and 

s = 2, 6, 12, 20 . . . (Feet). 

2. From the formula for the velocity 

« = 10 + 3^ — f, we obtain by integration 

s= flOdt + r^tdt- Ctdt = lot + |e«-|, 

and on the contrary by differentiation p = S — 2t. 

Consequently, for 3 — 2 ^=0, i.e., for i=f seconds, the acceleration is 
and the velocity is a maximum (v = 12J), and for 10 + 3 ^ — ^^ = 0, i.e., for 

3 + 7 

^=|4.VlO + f= — - — = 5 the velocity is = and the space is a masd- 

2 

mum. 

For ^ = 0, 1, 2, 3, 4, 5, 6 seconds we have 
p= S, 1, _ 1, _ 3, - 5, - 7, - 9 feet, 
V = 10, 12, 12, 10, 6, 0,-8 feet, 
8 = 0, 111, 23i, 34^, 42|, 45|, 42 feet. 

3. For the motion expressed by the formula p = — ju s, in which fi des- 
ignates a constant coefficient, we have 



v^ — c^ 



2 



= / pd8 = — fi / sds= —^, or v"^ =:c^ — fis"^; 



7^2 _ ^2 

whence v = vc^ — jus'^ and s =y • 

Wehavealso dt=:^= '^» ^ "^ ' 



V ^q2 ,. o2 C / /. ,/— \2 



- H 8 

's V/U 



]/i - ki±S 



V c I du 



v^/i-fi^y ^"^i-"'- 



when we put fJlif = u ; and it follows that (see Art. 26, V., of the Introduc- 

c 
tion to the Calculus). 

t = —— sin. -^ u~ ——sin.-^ LlJi^ and 

_ G 

* ~" ~p sin. (t V fi^, as well as 

d 8 , ,- ^ 

V = -jz. = c COS. {t V fj) and 

p = ^— = — c '/~fi sin. {t V fi). 

When the motion begins we have, for < = 0, « = 0, d = c and ^ = 0, 
and afterwards for 



21. J SIMPLE MOTION. 121 



< V^ = TT, or ^ = -/=j « = 0, « = — c and^ = 0, for 

V fJ, 

« V^ = f TT, or « = 2~7=' ^-~7J=''^^^ ^^^^ = ^ V/^> and for 

2 77 

i V /z = 2 TT, or ^ = /- , s = 0, -y = c and^ =0. 

The moving point has therefore a vibratory motion upon both sides of the 
fixed point of beginning, to which it returns every time that it has de- 
scribed, with a velocity which gradually increases from to « = db c, the 

c 
space s = zb . 

§21. Mean Velocity. — The velocity cji= —, which we find 

when we divide the space described during a certain time, e.g., 
during the period of a periodic motion, by the time itself, differs 

from the velocity v = — (y^) for an instant or during the ele- 
ment of time T (d t). We call the former the mean velocity (Fr. 
Vitesse moyenne, Ger. mittlere Geschwindigkeit), and we can con- 
sider it as the velocity that a body must have, to describe uniformly 
in a certain time (t) the space (s) which it really does describe with a 
variable motion in the same time. When the motion is uniformly 
variable the mean velocity is equal to the half sum of the initial and 
of the final velocity, for according to § 13 the space is equal to this 

(c -\- v\ 
— - — I multiplied by the time (t). 

In general, the mean velocity is (according to § 19) Ci = 

— ^ — — -, in which Vi, v>2, . . . v„ denote the velocities corre- 

n 

spending to equal and very small intervals of time. 

Example. — While a crank is turned uniformly in a circle TJ M N'y 
Fig. 51, the load @ attached to it, e.g., the piston of an 
air or water pump, etc., moves with a variable motion up 
and down ; the velocity of this load is at the highest and 
lowest points CTand a minimum, and equal to zero, and 
at half the height at Jf andiVa maximum, and equal to the 
velocity of the crank. Within a half revolution the mean ve- 
locity is equal to the whole height of ascent, i.e., the diam- 
eter £7" Oof the circle in which the crank revolves, divided 
by the time of a half revolution. If we put the radius of the 
circle in which the crank revolves, C U = C = r^ that 




GENERAL PRINCIPLES OF MECHANICS. 



[§22,23, 



is, its diameter = 2 r, and the time equal to % it follows that the mean 
velocity c^ = — . The crank in the same time describes a half circle 

TT r, and its velocity is c = — , and therefore the mean velocity of the load 



2t 



^'~ 7T '- d^ui 



c is 0,6366 times as great as the constant velocity c of 



the crank. 



§ 22. Graphical Representation of the Formulas of Mo- 
tion, The laws of motion which have been found in the foregoing 
paragraphs can be expressed by geometrical figures, or, as we say, 
graphically represented. Graphical representations, as they ren- 
der the conception of the formula more easy, assist the mem- 
ory, protect us from many errors, and serve also directly for 
the determination of quantities which may be required, are 
of the greatest use in mechanics. In uniform motion, the space 

(s) is the product (c t) of the velocity and 
the time, and in Geometry the area of a rect- 
angle is equal to the product of the base by 
the altitude ; we can therefore represent the 
space described (s) by a rectangle A B C D, 
Fig. 52, whose base ^ ^ is the time t, and 
whose altitude A D ~ B G i^ the velocity c, 
provided the time and the velocity are expressed by similar units 
of length, that is, if the second and the foot are represented by 
one and the same line. 




Fig. 53. 



§ 23, While in uniform motion the velocity {M N) at any mo- 
ment {A M) is the same, in variable motion it is different for each 
instant ; therefore this motion can only be represented by a four- 
sided figure, A B C D, Fig. 53, the base 
of which A B, denotes the time {t), the 
other boundaries being the three lines, 
A D, B C, and C D, The first two 
of these lines denote the initial and final 
velocities, and the last one is determined 
by the extremities {N) of the different lines 
representing the velocities corresponding 
to the intermediate times {M). Accord- 
ing to the nature of the variable motion in question, the fourth 
line (7 Z> is straight or curved, rises or sinks from its origin, and is 




2i.] 



SIMPLE MOTION. 



123 



concave or convex towards the base. In every case, however, the 
area of this figure is equal to the space (s) described ; for every sur- 
face ABC D, Fig. 53, can be divided into a series of small 
strips M P N, which may be considered as rectangles, and the area 
of each of which is a product of the base [M 0) and the corresponding 
altitude ( M N) or (0 P), and in like manner the space described 
in a certain time is composed of small portions, each one of which 
is a product of an element of time and the velocity of the body 
during that instant. The figure also shows the difference between 
the measure of the velocity and the space actually described in the 
following unit of time. The rectangle M L, above the base 
M H — unity (1) = ■?; . 1 is the measure of the velocity M, and on 
the contrary, the surface M K above the same base represents the 
space actually described. In the same way the rectangle A F over 
A I — unity is the measure of the initial velocity A D = c, and the 
surface A E that of the space actually described in the first second. 

§ 24. In uniformly variable motion the increase or decrease v—c 
of the velocity (= ^ ^, § 13) is proportional to the time {t). If in 
Fig. 54 and Fig. 55 we draw the line D E parallel to the base A B, 
we cut off from the lines B C and M N, which represent the velo- 





cities, the equal portions B E and M 0, which are equal to the 
line A D representing the initial velocity, there remain the pieces 
C E and N 0, which represent the increase or decrease in velocity ; 
for these we have from what precedes the proportion 

N OiC E = D 0:D E. 

Such a proportion requires that JV, as well as every point of the 
line C D, shall be upon the straight line uniting C and D, or that 
the line CD, which limits the velocities M N, shall be straight. Con- 
sequently the space described in uniformly accelerated or retarded 
motion can be represented by the area of a Trapezoid A B C D, 



124 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 35, 26. 



whose altitude ^ ^ is the time (t) and whose two parallel bases 
A D and B C are the initial and final velocity. The formula found 

in § 13 5 = — ^ — . t corresponds exactly to this figure. For uni- 

formly accelerated motion the fourth side D C rises from the point 
of origin, and for uniformly retarded motion this line descends 
from the same point. When the uniformly accelerated motion be- 
gins with a Telocity equal to zero, the trapezoid becomes a trian- 
gle, whose area i^^BC.AB = ^vt. 

§ 25. The mean velocity of a variable motion is the quotient of 
the space divided by the time ; it gives, when multiplied by the 
time, the space, and can be considered as the altitude A F — 
B E oi the rectangle A B E F, Fig. 56, the base of which A B is 
equal to the time t, and the area of which is equal to that of the 
four-sided figure A B C N D, which measures the space described. 
The mean velocity is found by changing the four-sided figure A B 
C N D into an equally long rectangle A B E F. Its determina- 
tion is especially important for periodic motion, which occurs in 
almost aU machines. The law of this motion is represented by 
the serpentine line G D E F G, Fig. 57. If the right line L M, 





drawn parallel to A B, cuts off the same space as the serpentine 
line, then L M\^ also the axis oi G D E F G, and the distance A L 
= B M between the two parallels A B and L Mis the mean ve- 
locity of the periodic motion, and, on the contrary, AG, E, B G, 
etc., are the maximum, and N D and P F the minimum velocities 
of a period A 0, B, etc. 

§ 26. The acceleration or the continuous increase of velocity in a 
second can easily be determined from the figure. In uniformly 
accelerated motion it is constant, and is therefore the difference 
F Q, Fig. 68 and Fig. 59, between the two velocities OP and M N, 



§26.] 



SIMPLE MOTION. 



125 



one of which corresponds to a time {M 0) one second greater than 
the other. If the motion is variable^ but not uniformly, and the hne 




Fig. 59. 




M O 



M O 



of velocity C D therefore a curve, the acceleration at every instant 
is different, and consequently it is not really the difference P Q oi 
the velocities P and M N = Q, Figs. 60 and 61, which are 
"those at times differing one second M from each other, but it 




Fig. 61. 




MO B 



is the increase R of the velocity M N, which would take place, 
if from the instant M the motion became a uniformly accelerated 
one, that is if the curve N P C became a straight line N E, But 
the tangent iV^ ^ is the line in which a curve D N would prolong 
itself, if from a certain point {N), its direction remained unchanged ; 
the new line of velocity coincides with the tangent, and the perpen- 
dicular R which reaches to this line is the velocity which would 
have existed at the end of a second, if at the beginning of the same 
the motion had become a uniformly accelerated one, and therefore 
the difference R Q between this velocity and the initial one {M N) 
is the acceleration for the instant which corresponds to the point 
M in the time line A B. We can also of course consider the time 
and the accelerations as the co-ordinates of a curve, in which case 
the velocities are represented by surfaces. 



126 GENERAL PRINCIPLES OF MECHANICS. [§27,28. 



CHAPTER II. 

COMPOUND MOTION. 

§ 27. Composition of Motion. — The same body can possess, 
at the same time, two or more motions ; every (relative) motion is 
composed of the motion within a certain space, and of the motion 
of this space within or in relation to another space. Every point 
on the earth possesses already two motions; for it revolves once 
every day around the earth's axis, and with the earth once a year 
around the sun. A person moving on a ship has two motions in 
relation to the shore, his own motion proper and that of the ship ; 
the water which flows out of an opening in the side or in the bot- 
tom of a vessel carried upon a wagon has two motions, that from 
the vessel, and that with the vessel, etc. 

Hence we distinguish simple and compound motion. The rec- 
tilinear motions of which other rectilinear or curvilinear motions 
are composed (Fr. composes, G-er. zusammengesetzt), or of which 
we can imagine them to be composed, are simple motions (Fr. sim- 
ple, Ger. einfach). How several simple motions can be united so 
as to form a compound one, and how the decomposition of a com- 
pound motion into several simple ones is accomplished, will be 
shown in what follows. 

§ 28. If the simple motions take place in the same straight line, 
their sum or difference gives the resulting compound motion, the 
former when the motions are in the same direction, and the latter 
when the motions are in opposite directions. The correctness of 
this proposition becomes evident, when we combine the spaces de- 
scribed in the same time by virtue of the simple motions. The 
spaces Ci t and c^ t described in the same time correspond to uni- 
form motions whose velocities are Cx and Cs, and if these motions 
are in the same direction the space described in t seconds is 

and consequently the resulting velocity of the compound motion is 
the sum of the velocities of the simple motions. When the mo- 
tions are in contrary directions, we have 



^ 29, 30.] COMPOUND MOTION. 127 

S — Cit — C-jt — {Ci — c^ t, 

and tlie resulting yelocity is equal to the difference of the simple 
velocities. 

ExAJiPLE. — 1. A loerson, walking upon the deck of a ship with a velo- 
city of 4 feet in the direction of the motion of the latter, aj)pears to people 
on shore, when the ship moves with a velocity of 6 feet, to pass by with a 
velocity of 4 + 6 = 10 feet. 

2. The water discharged from an opening in the side of a vessel with a 
velocity of 25 feet, while it is moved simultaneously with the vessel in the 
opposite direction with a velocity of 10 feet, has in reference to the other 
objects which are at rest a velocity of only 25 — 10 = 15 feet. 

§ 29. The same relations also obtain for variable motion. If 
the same body has, besides the initial velocities Cx and c^, the con- 
stant accelerations^! andj^s, the corresponding spaces are c^ t, Cg t, 
i i^i f, A 7^2 f, and if the velocities and the accelerations have the 
same directions, the total space described in virtue of the compo- 
nent motions is 

f 

f 
If we put d -\- d = c and pi + 2^c^ = p, we obtain s = c t -\- p -, 

whence it follows that not only the sum of the component 
velocities gives the velocity of the resulting or compound 
motion, but also that the sum of the accelerations of the 
simple motions gives its acceleration. 

Example. — A body upon the moon has imparted to it by the 
moon an acceleration p^ = 5,15 feet, and from the earth an ac- 
celeration ^3 = 0,01 feet. Therefore, a body A, Fig. 62, beyond 
the moon M and the earth E^ falls towards the centre of the 
moon with an acceleration of 5,16 feet, and a body B between M 
and E with an acceleration of 5,14 feet. 

§ 30. Parallelogram of Motions, — If a body possesses at the 
same time two motions which differ from each other in direction, 
it takes a direction which lies between those of the two motions, 
and if these motions are of different kinds, e.g., if one is uniform 
and the other variable, the direction changes at every point, and 
the motion is curvilinear. 

We find the point 0, Fig. 63, which a body moving at the same 
time in the direction A X and A Y, occupies at the end of a cer- 




128 



GENERAL PRINCIPLES OF MECHANICS. 



[§31. 



Pig. 63. 



tain time (t) by seeking the fourth corner of the parallelogram A 
M N, determined by the spaces A M ~ x and A N — 'y, de- 
scribed simultaneously, and by the angle X A F which the direc- 
tions of motion form with one another. 
We can convince ourselves of the correct- 
ness of this proceeding by supposing the 
spaces X and y described not simultane- 
ously, but one after the other. By virtue 
of one motion the body describes the space 
A M ^=^ X, and by virtue of the other from 
M in the direction A Y, that is on a line 
M parallel to A Y, the space A N — y. 
If we make Jf = ^ iV, we obtain in 
the position of the body which corresponds to the two motions x 
and y, and which, according to this construction, is the fourth cor- 
ner of the parallelogram. We can also imagine the space AM=- 
X to be described in a line A X, which with all its points moves 
forward in the direction A Y, and therefore carries M parallel to 
A Y and causes this point to describe the path M = A N = y. 




§ 31. Parallelogram of Velocities. — If the two motions in 
the directions A X and A Y take place uniformly with the ve- 
locities Ci and Co, the spaces described in a certain time t are x = 

Cx t and y = c^t, and their ratio ^ = — is the same for all times, 



X 



Fig. 64. 



a peculiarity which is possessed only by 
the right line A 0, Fig. 64. It follows 
therefore that the direction of the com- 
pound motion is always a straight 
line. If we construct with the veloci- 
ties A B — Ci and A C = c<i the paral- 
lelogram A B C Dy its fourth corner D 
gives the point where the body is at 
the end of the first second, but since 
the resulting motion is rectilinear, it 
follows that it takes place in the direction of the diagonal of the 
parallelogram constructed with the velocities. If we designate by 
s the space A really described in the time t, we have from the 
similarity of the triangles AM and ABB 




§32.] 



COMPOUND MOTION. 



129 



s A D 
= -j-^f whence it follows that this space 



X 



s = 



x.AD Cit .A D 



= AD.t. 



AB c. 

According to the last equation the space described in the di- 
agonal is proportional to the time {t), and therefore the compound 
motion is itself uniform and its velocity c equal to A D, 

Therefore the diagonal of a ]}araUelogram, constructed with tivo 
velocities and with the angle inclosed hy them, gives the direction 
and magnitude of the velocity, ivith which the resulting motion actu- 
ally tahes place. This parallelogram is called the parallelogram 
of velocities (Fr. parallelogramme de vitesse, Ger. Parallelogram 
der Geschwindigkeiten) ; the simple velocities are called compo- 
nents (Fr. composantes, Ger. Seitengeschwindigkeiten), and the 
compound velocity the resultant (Fr. resultante, Ger. die resulti- 
rende or mittlere). 

§ 32- By employing trigonometrical formulas, the direction 

and magnitude of the resulting veloc- 
ity can be found by calculating one 
of the equal triangles, e.g., A B D, 
of which the parallelogram of velocities 
is composed, by which we obtain the re- 
sulting velocity A D = c in terms of 
the components A B = Ci and A C = 
d and of the angle included between 
them B A C = a. 

For we obtain c by the formula 
c = Vci^ + Cs* + 2 Ci c<2 cos. a, 
and the angle B A D = (f), which the resultant makes with the 

velocity c^, by the formula sin. </> == ^, or 



Fig. 65. 




d . sm. a 

tang, (h — , 

Ci + Ci cos. a 

We have also 



or cotang. (j) = cotang. a + 



Cx 



d sm. a 



— Co 



a 



If the velocities c, and Cc, are equal to each other, the parallelo- 
gi'am is a Rhombus, and in consequence of the diagonals being at 
right angles to each other, we have more simply 

c = 2ci COS. ^ a and = ^ a. 



130 GENERAL PRINCIPLES OF MECHANICS. [§ 8a 

If the velocities are at right angles, we have also more simply 

^2 



c — ^/cx + c^' and tang. = — . 

^1 

Example. — 1. The water discharged from a vessel or from a machine 

has a velocity c^ = 25 feet, while the vessel itself is 'moved with a velocity 

(?3 = 19 feet in a direction, which forms with that of the water an angle 

a" = 130'\ What is the direction of the resultant or absolute velocity 

of the water ? 

c = V25'^ + 19'^ + 2 . 25 ri9 cos. 180" = V625 + 361 - 50 . 19 7cos. 50'. 

= V986- 950 'cos/5¥= V986 - 610,7 =V375,8 = 19,37 feet 

is the required resulting velocity. 

1 9 s/w 1 SO" 
Further, si?i. <f> = — ' = 0,9808 sin. 50" = 0,7513, hence the 

angle formed by the direction of the resultant with that of the velocity Cj is 
(p = 48" 42\ and the angle formed by it with the direction of the motion 
of the vessel is a — ^ = 81" 18\ 

2. If the foregoing velocities were at right angles to each other, we 

would have COS. a = cos. 90'' = 0, and therefore the resulting velocity c=V986 
= 31,40 feet, and also tang. (p = ^^ = 0,76, hence the angle formed by it with 
the first velocity is ^ = 37" 14' . 

§ 33. We can also consider every velocity to be composed of 

two components, and therefore under 
""• certain conditions can decomDose it 

into such components. If, for example, 
the angles D A X — (p, and DAY 
— ijj, Fig. 66, which the required 
velocities form with the resultant 
A D = c, are given, we draw through 
the extremity i) of the line represent- 
ing c other lines parallel to the di- 
rections A X and A Y : the points of 
intersection B and D cut off the ve- 
locities sought, and we have 

A B = Ci and A C = c^. 
Trigonometry gives these velocities by the formulas 

c sin. lb c sin. 

sin. {(f) + xpy ^ sin. (0 + V^)* 

* 

Generally, in the application of these formulas, the two velocities 
are at right angles to each other, and 

(f) -{- if) z= 90", sin. (0 4- V) = 1, whence 
Ci = c COS. and a, = c sin. 0. 




§ 34.] 



COMPOUIND MOTION. 



131 



We can also determine, when one component (ci) and its angle 
of direction {(f)) are given, the magnitude and direction of the 
other. Finally, if the three velocities c, Ci and Ca are given, we can 
determine their angles of direction by the same method that we 
employ to find the angles of a triangle, when three sides are given. 

Example. — If the velocity c = 10 feet is to be decomposed into two 
components whose directions form with that of c the angles (p = 65" and 
i{j = 70°, we have 



10 sin. 70' _ 9^397 
sin. 45 



c — 
^ sin. 135" 



-J = 13,39 feet and Cg = 



10 sin. 65" 9,063 



sin. 135" 0,7071 



: 12,81 feet. 



§ 34. Composition and Decomposition of Velocities. — 

By repeated use of the parallelogram of velocities, any number of 
velocities can be combined so as to give a single resultant. 
The construction of the parallelogram A B D C (Fig. 67) gives the 
resultant A D oi^ c^ and Co, the construction of the parallelogram 
A D F U gives the resultant of ^ i> and A U = Cs, and from the 
construction of the parallelogram A F ff G we obtain the result- 
ant A R = c of A Fand A G = c^, or that of Ci, Cs, C3 and c^. 

The most simple manner of resolving this problem is by the 
construction of a polygon A B D F H, whose sides A B, B D, D F 
^nd F H are parallel and equal to the given velocities c^, c^, c^ and 
C4, and whose last side is always equal to the resulting velocity. 




Fig. 68. 




In case the velocities do not lie in the same plane, the re- 
sultant can also be found by repeated application of the paral- 
lelogram of velocities. The resultant A F = c (Fig. 68) of three 
velocities A B = c^, A O — d and A E — c.^, not in the same 
plane, is the diagonal of a parallelopipedon whose sides are equal 




132 GENERAL PRINCIPLES OF MECHANICS. [§ 35, 3S. 

to the velocities. We often employ for this reason the term parol- 

lelopipedon of velocities, 

§ 35. Composition of Accelerations. — By the composition 

of two uniformly accelerated motions, beginning with a velocity = 

0, we obtain also a uniformly accelerated motion in a straight line. 

If we designate the accelerations of the motions in the directions 

A Xand^ Y(F\g. 69) by^i and^^g? the spaces described during the 

time t are 
Fig. 69. ^ p 

and their ratio is 

X _ p^ f _ Pi 
y~P^f~ p^ 
which is entirely independent of the 
time, therefore the path ^ is a 
straight line. If we make A B = p^, 
and B D — A C = p^^we obtain a parallelogram A B D C, and 
we have 

4-^ = 4_^^kll£ = 1 t\ whence AO=^hAD.f 
A D A B pi ^ 

According to this equation the space A of the compound motion 
is proportional to the square of the time ; the motion itself is there- 
fore uniformly accelerated, and its acceleration is the diagonal A D 
of the parallelogram constructed with the two simple accelera- 
tions. 

We see, therefore, that we can combine several accelerations so 
as to form a single one, or decompose a single one into several 
others by means of \h.Q p)arallelogram of accelerations (Fr. parallel- 
ogramme des accelerations, Ger. Parallelogram der Accelerationen) 
according to exactly the same rules as we perform the composition 
and decomposition of velocities by means of the parallelogram of 
velocities. 

§ 36. Composition of Velocities and Accelerations. — 

By the combination of a uniform motion with a uniformly ac- 
celerated one we obtain, when the directions of the two motions 
do not coincide, a motion which is completely irregular. If during 
a certain time t, by virtue of the velocity c, the space 

A N = y — ct 



§36] 



COMPOUND MOTION. 






is described in the direction A Y, Fig. 70, and if during the same 
time, by virtue of a constant acceleration, the space 

J9f 



A M 



X 



2 



is described in the direction A X ai right angles to the former, 
then the body will be in the corner of the parallelogram con- 



structed with y = c t and x 



pf 



. By the aid of these formulas, 

it is true, we can find the position of the body for any given time, 

but these positions do not lie in the same straight line ; for if we 

1/ 
substitute the value of ^^ = - taken from the first equation, m the 

c 

second we obtain the equation of the path 

^ ~ 2 c^- 
According to this formula the space (x) described in one direction 
varies, not as the space, but as the square (y^) of the space described 

Fm. 70. 





in the other direction, and the path of the body is therefore not a 
straight line, but a certain curve known in Geometry as the parab- 
ola (Fr. parabole, Ger. Parabel). 

Remakk,— Let ABC, Fig. 71, be a cone with a circular base AEBF, 
and B EF a section of the same parallel to the side B C and at right an- 
gles to the section A B (7, and let OP NQ be a second section parallel to 
the base and therefore circular. Further, let E F be the line of intersec- 
tion between the base and the first section, and finally, let us suppose the 
parallel diameters A B and P ^ to be drawn in the triangular section ABC 
and the axis B Gin the section B E F. Then for the half chord MN 



M we have the equation M N'' = PM. MQ; but MQ= G B and for 



134 



GENERAL PRINCIPLES OF MECHANICS. 



[§37. 



P Mwe have the proportion PM: 1)M= A G : I) G, whence 

DM. A G 



MN' =BG 



B G 



But we have also G B^=B G.A G; whence, dividing the first equation 
by the second, 



Wg 



G E^ 



The 2^ortionn cut off from the axis {abscissas) are as the squares of the cor- 
responding perpendiculars {Ordinates). This law coincides exactly with 
the law of motion just found ; the motion takes place then in a curved 
line D N E^ which is one of the conic sections. For the construction, po- 
sition of the tangent, and other properties of the parabola, see the Inge- 
nieur, page 175, etc. 

§ 37. Parabolic Motion. — In order thoroughly to under- 
stand the motion produced by the combination of velocity and 
acceleration, we must be able to give for any time {t) the direction, 
velocity, and the space descrihed. The velocity parallel to ^ 1^ is 
constant and = c, and that parallel to A Xis variable and = pt', 

if we construct with these ve- 
^^^- ^^' ' locities e = c and P = ;j 2f 

the parallelogram P R Q, 
Fig. 72, we obtain in the di- 
agonal R the mean velocity, 
or that with which the body in 
describes the parabolic path 
A U. This velocity itself is 

V — V c' + {^j ty. 

R gives also the tangent 
or the direction in which the 
body moves for an instant ; con- 
sequently, for the angle P R ~ X T =0, which the same 
makes with the direction (axis) A X oi the second motion, we 
have the following formula 

tang. = g-| = j^- 

Finally, to obtain the space described or the arc of the curve 
A = s,^Q can employ the formula a — v r [^ 19), by the aid of 
which we can calculate the small portions which we can consider 
as elements. The calculus also gives a complicated formula for the 
computation of an arc of a parabola. 




Fig. 73. 



§38.] COMPOUND MOTION. I35 

^ § 38. We have previously supposed that the primitive directions 
of motion were at right angles to each other, and we must now 
consider the case, when the direction of the acceleration makes any 
arbitrary angle with that of the velocity. If the velocity oi 
the body in the direction A Yx (Fig. 73) is c, iind if, in the 

direction A X^ which forms 
an angle X, A Y^ = a with 
the former, the acceleration 
is p, A is no longer the ver- 
tex, and A X^ no longer 
the axis, but only the di- 
rection of the axis of the 
parabola. The vertex of the 
parabola is situated at a 
point whose co-ordinates, in 
reference to the point of be- 
ginning of the motion, are C B = a and B A — b, of which the former 
lies in the axis of the parabola and the latter is at right angles to 
it. The velocity A D = c is composed of the two components 
A F = c sin. a and A E— c cos. a. The first of these is constant, 
and the latter is variable, and always equal to the variable velocity 
7; t, provided that the body requires the time t to pass from the 
vertex C to the real point of beginning. 
Hence we have 

, , , ^' • COS. a 

c COS. a = JO t, whence t = 



c 


T 






N 


B 


a ^'^""■^^^;^ 


J^ 


,,F 








b 


E^ 


K 

t 






IW 


\ 


K 


3 


c 











w. 



^x 



and therefore 



P 



1) GB = a 



2 



& cos?' a 



%p 



and 



2) B A = h = c sin. a . t 



c^ sin. a cos. a & sin. 2 a 



p 2 p 

If we have determined by these distances the vertex C of the 
parabola, starting from this point we can, for any given time, de- 
termine the position of the body. Besides, if we put C M^^ x and 
M — y, the general formula 



X = 



p y 



ovy 



c sm. 



a\/ - 



2x 



holds good. 



2 c' sin? a " ' p 

Remark. — One of the most important applications of the theory of par- 
abolic motion, just discussed, is to the motion of projectiles. A body pro- 
iected in an inclined direction either upward or downward would describe, 
in virlue of its initial velocity c and of the acceleration of gravity (j7 = 32 . 2 
fcot), an arc of a parabola, if the resistance of the air were done away with, 



136 



GENERAL PRINCIPLES OF MECHANICS. 



[§39. 



or if its motion took place in vacuo. If tlie velocity of projection is not very- 
great and if the body is very heavy compared with its volume, the diver- 
gence of the body from a parabolic path is small enough to be neglected. 
The most perfect parabolic trajectories are those described by jets of water 
issuing from vessels, fire-engines, etc. Bodies shot from guns, etc., e.g., 
musket balls, describe, in consequence of the great resistance of the air, 
paths which differ very sensibly from a parabola. 

§ 39. Motion of Projectiles.— A body projected in the di- 
rection A ]r at an angle 
Fi<^' '^^ of elevation YAD^a, 

Fig. 74, ascends to a cer- 
tain height B C, which 
is called the lieiglit of 
projection (Fr. hauteur 
du jet, Ger. Wurfhohe), 
and it reaches the hori- 
zontal plane from w^hich 
it started in ^, at a dis- 
tance A D from it, which 
is called the range of 
projection (Fr. ampli- 
tude du jet, Ger. Wurfweite). 

From the velocity c, the acceleration g and the angle of eleva- 
tion, we obtain, according to § 38, when we replace p hj g and 

a" by 90" -f a% or cos. a by sin. a, etc. 

c^ sin^ cb 
the height of projection C B = a — — ^r-^ — and 




half the range of projection A B — h 



& sin. 2 a 



From the last formula we see that the range of projection is a 

maximum for sin. 2 a = 1, ot 2 a = 90°, that is for a = 45". A 

body projected at an angle of elevation of 45° attains the greatest 

range of projection. 

We have also 

gb' 



a = 



2 & cos.^ a' 



and for a point in the path of the projectile for which G M = x 

and M = y, 

/v ^— ^ ^ 

2 & cos."" a' 

or when its position is given by the co-ordinates A .N 

N = yx, since in that case 



X. 



and 



§39.] COMPOUND MOTION. 137 

X = CM^BC— NO — a — y^ and 

y = MO~AB-AN=h — x^, we have 

a - y, =^ ^ 



2 c' cos.^ a 
whence 



(j{})- x,y . g ¥ 

y^ — a — "--^ i"^, or since a — --^ ^ 

^ % c COS. a 2 c cos.^ a 



gx,' 



V\ = X\ tang. o. — — .^ ^ . 
^ ^ 2 c COS. a 



g x{ 



Substituting in the equation y^ = x^ tang, a — f ' \ , for 
-; — , the value 1 + tanq.^ a, and resolving the same in reference 



COS.' a 



to tang, a, we obtain the following expression for the angle of eleva- 
tion (a), required to reach a point given by the co-ordinates x^ 
and y„ 

tang, a = -^-- ± /(-ZTrTTT^ZlA. 

^ q x^ \g xj \ g x^ I 

(' & ^ 2 & y 
1 = 1 H -74"^ or d^ — 2 g y^ c' = g^ x-^^, then we have 
g Xxl g Xi 

c =^^g (yi + ^a^i' i- yi') and 
tanq. a = . 

Smaller values of c make tang, a imaginary, and larger values of c 
give two values for tang, a; in the first case the point cannot be 
attained, and in the second case it would be attained either in the 
rise or in the fall of the projectile. 

Example. — 1. A jet of water rises with a velocity of 20 feet at an angle 

of 66°. The height due to the velocity is h = 0,0155 . 20' = 6,2 feet, and 

the jet ascends to a height a = h sin."^ a = 6,2. (szVi. 66°)' = 5,17 feet, the 

range of the jet is 2 5 = 2 . 6,2 sin. 132° = 2 . 6,2 sin. 48 = 9,21 feet. The 

time, which each particle of water requires to describe the entire arc A CD 

r..-, u 1 • ^ 2 c sin. a 2.20sz7i. 66° ^^, , ^, , 

of the parabola, is t = = --^ =1,14 seconds. The height 

corresponding to the horizontal distance A J^=:Xj^ =3 feet is 

V - 3 tano 66° — ?^^ ' ^ - 6 7-H ^'^^^^^ 

^, _ 3 . tang. 66 - g 400 . ^,7(66^ " ^'^'^^ " pesTS 

= 6,738 - 2,189 = 4,549 feet. 
2. A jet of water discharged from a horizontal tube has, for a height If 
feet, a range of 5^ feet ; how great is its velocity ? 



130 GENERAL PRINCIPLES OF MECHANICS. [§ 40, 

g y^ y"^ y"^ 

From the formula x = ^^—^ = a-t- ^^ deduce h =:^ , in which we must 

2 c^ 4:7i 4a? 

substitute x = 1,75 and y = 5,35, and thus we obtain h = "^ ^k — ^^^S? 

feet and the corresponding velocity c = 15,92 feet. 

§ 40. Jets of Water. — The peculiarities of the motion of jets 

of water are explained and shown in what follows. From what 

precedes we have 

a x" 11 4- (tanq. a)'] , 
y -=z X tang, a — --\ — - — '-^ and 

for the equations of the parabolas formed by the paths of two as= 
cending jets of water whose velocities c are the same, and whose 
angles of elevation a and a^ are ditferent. If we put x^ = x and 
subtract these equations from one another, we obtain 

y — y^ — X (tang, a — tang, a^ — '^ , \_{tang. a)- — {tang. a^)'~\ 



ij 



% & 



= X {tang, a — tang, a^) (1 — ^'^ {tang, a + ta}2g. a-^)). 

If we assume that the two streams have nearly the same anoie 
of elevation and require the two parabolas to have a point in com- 
mon, we must put yi = y and consequently we have 

X {tang, a — tang, ^i) (l — ^ {ta7ig. a + tang, a-^)) ~ 0, or 

o X 

^-^ {tang, a + tang, a^ = 1, 

or, since we can put aj = a we have simply 

a X tanq. a _ i , c' 

~ — = 1, whence tanq. a — ■ — . 

c- ^ g X 

Substituting this value in the equation 

y ~ X tang, a — |-— ^ [1 + {tang, a)'], 
we obtain the equation 

^^ ~~g ~ 2~^'\ '^ f^'l ~^ 2g~ ¥? 
of the curve D F S P D, Yig. 75, which passes through the neigh- 
boring points, in which every two parabolas starting from the same 
point A at different angles cut each other, and which, therefore, 
touches or envelops the whole system of parabolas A C D, A R, 
etc. 



40.] 



COMPOUND MOTION. 



139 



The heiffht to which a vertical net of water rises is A 8 = ^r-, 
and the range of projection of a jet ^ CD rising at an angle of 

Fia. 75 




^6° is AD = 2 . 



c^ sin. 2 a 



z=z 2 . 



2 A S. 



^g ^g 

If we transfer the origin of co-ordinates from A to 8, re- 
placing the co-ordinates A N= x and N P — ^^ by the co-ordinates 
8 IT = tc and UP = v, we have 

y= A 8- 8 U= f'-timdx:= A. iV^ UP = v, 

-^ ^g 



and the equation 



y 



g ^ • 



2 (J 2 & 



is thus transformed into 



g ^ 
2 c 



2e 
g 



u. 



This ecjAiation is that of the common parabola whose parameter 

2c^ ' 

is =r — = ^ A 8, and therefore the envelope D P 8 P D of all 

g 



140 



GENERAL PRINCIPLES OF MECHANICS. 



li 40. 



the jets of water rising from the point A is a common parabola, 

whose vertex is S and whose axis is ^iS' ^. 



Fig. 76. 




A bunch of jets rising from A m all directions would be envel- 
oped by the paraboloid generated by the revolution of the envelope 
D P S P D around A S. If t is the time in which a body rising in 
a parabola describes the arc A 0, Fig. 76, the co-ordinates of which 
are A M =^ x and M = y, we have 

2 

y + hg^ 



X 



c t COS, a and y = c t sm. a — ^—-, whence 



X 



COS. a = 



ana stn, o. = 



c t c t 

Substituting these values for cos. a and sin. a in the well-known 
trigonometrical formula {cos. ay + (sin. ay = 1, we obtain the fol- 
lowing formula 



^' , (y + igiT. 



+ 



1, or x' + {y + ig ty = c' t\ 



(cty ' (cty 

If from a point A, Fig. 76, bodies be projected at the same mo- 
ment and in the same vertical plane at different angles of eleva- 



§41] COMPOUND MOTION. 14X 

tion, the positions that they occupy after the lapse of a certain 
time (t) are determined by the last equation, which is that of a circle 
whose radius is r = ct and whose centre is situated vertically below 
^ at a distance a = } g f, and which can therefore be written in 

the following form, 

x" + (y + ay — r". 

The circumference of this circle would therefore be reached at the 
same moment by all the elementary jets A G D, A P, A L S».., 
rising at the same moment from the point A. 

If in the formula ti = we substitute a = 45° and x = 

c COS. a 

A B = 7z—, we obtain ti = r-^ = — VT, hence the time re- 

2 g 2 g cos. 45° g 

-quired to describe the whole arc of the parabola A CD is t = 

2 ti = — V2, and the radius of the circle D L D, which is reached 
9 

simultaneously by the different elements of the water, is 

2 2 2 

KD = r ^ ct = -^ V2 = ^ V8 = 2,828 ^ == 2,828 . ATS, and 
the distance of the centre K from A is 



-^ g 2g 

If we divide D K in 4, and A K'vcl 16 equal parts, we can, since 
T is proportional to t and a to f , from the points of division 1, 4, 9 
in A K, describe other circles with the radii \ D K, i D K, and 
^ D K, which cut off the parabolic arcs described in the same time, 
E.G., the circle described from 1 with 1 a = \ D K, cuts off in the 
points a, a^, , the parabolic paths A a, Aa^ , described simul- 
taneously, and the circle described from 4 with 4t 13 = ^ D ^cuts off 

in the points (3, I3i the parabolic arcs A 13, A (3^, etc., which are 

also simultaneously described. 

If these circles be revolved about the vertical axis ^Z, they de- 
scribe spherical surfaces which bound the parabolic paths described 
simultaneously, when the jets are projected all around A at all 
angles of elevation. 

§ 41. Curvilinear Motion in General — By the combination 
of several velocities and several constant accelerations, we obtain 
also a parabolic motion, for not only the velocities but also the ac- 
celerations can be combined so as to form a single resultant ; the 



142 



GENERAL PRINCIPLES OF MECHANICS. 



[§43. 



problem is then tlie same as if there were one velocity and one 
acceleration, i.e., as if there were but one uniform and one uni- 
formly accelerated motion. 

If the accelerations are variable, they can be combined so as to 
give a resultant, as well as if they were constant, for we can con- 
sider them as constant during an infinitely small period of time (r), 
and the motion as uniformly accelerated during this time. The 
resulting acceleration is, it is true, like its components themselves, 
variable. If we combine this resulting acceleration with the given 
velocity, we obtain the small parabolic arc, in which the motion 
takes place during this instant. If we determine also for the follow- 
ing instant the velocity and the acceleration, we obtain another por- 
tion of an arc belonging to another parabola, and proceeding in the 
same manner, we obtain approximately the entire curve of the path. 

§ 42 We can consider every small arc of a curve as an arc of a 
circle. The circle to which this arc belongs is called the circle of 
curvcUure or oscillatory circle (Fr. cercle osculateur, G-er. Krum- 
mungskreis), and its radius is the radius of curvature (Fr. rayon de 
courbure, Ger. Kriimmungshalbmesser). The path of a body in 
motion can be considered as composed of such arcs of circles, and 

we can therefore deduce a 



Fig. 77. 




formula for its radii. 
A M (Fig. 77) 



X 



Let 

Pi 

be a very small space de- 
scribed in the direction A X 
with uniformly accelerated 
motion, A iV= y = t'ra very 
small space described uni- 
formly and the fourth cor- 
ner of the parallelogram con- 
structed with X and y, that 
is, the position that the body 
starting from A occupies at 
the end of the instant (r). 

Let us draw A C perpen- 
dicular to A y, and let us 
see from what point C in this line an arc of a circle can be de- 
scribed through A and 0. In consequence of the smallness of A 
we can consider not only CA, but also (7 P as perpendicular to 



^43.] COMPOUND MOTION. 143 

A F, SO that in the triangle iV^ P the angle N P can be 
treated as a right angle. The resolntion of this triangle gives 

P ^ ON sin. N P — A M sin. X A Y = ^ sm. a, 
and the tangent 

AP = AJV+ NP = VT -V- ^ cos. a= h +^ COS. aj r, 

can be pnt — vr, fov^ cos. a can be neglected in the presence of 

V, in consequence of the infinitely small factor r. Now, from the 
properties of the circle we know that A P' = P 0.{P + 2 C 0), 
or since P can be neglected in the presence of 2 (7 0, ^ P' = i^ (? 
. 2 (7 0; whence it follows that the radius of curyature is 

C A = G = r 



2 P pr'^ sin. a p sin. a 
In order to determine by construction the radius of curvature, 
we lay off upon the normal to the original direction of the motion 
A Y the normal acceleration, i.e., its normal component p sin. a 
= AD, and join the extremity E of the velocity A E = v to Dhj 
the right line D E, then we erect upon B E 2b perpendicular E G\ 
the* point of its intersection with the first normal is the centre of 
the osculatory circle of the point A. 

By inverting the last formula we obtain the normal accelera- 



v' 



tion n ^ p sin. a = — ; from which we see that it increases di- 
rectly as the square of the velocity, and inversely as the radius of 
curvature, or directly as the greatness of the curvature. 

Example. — The radius of curvature of the parabolic trajectory pro- 

duced by the acceleration of gravity is r = 0,031 —. , and for the vertex 

Sv?i» ct 

of this curve where a = 90°, and therefore sin. a = 1, it becomes r — 
0,031 c- feet. For a velocity c = 20 feet we obtain r = 12,4 feet ; the 
farther the body is distant from the vertex the smaller a becomes, and con- 
sequently the greater is the radius of curvature. 

§ 43. If the point A has described the elementary space A = 
G, its velocity has changed ; for the initial velocity v in the direc- 
tion A Yis now combined with the velocity /j> r acquired in the di- 
rection A X, and consequently from the parallelogram of velocities 
we have for the velocity Vx 

Vx = v^ + 2 V p r COS. a + p^ r'' = v^ -\- p r {2 V cos. a -\- p r). 
hutp r vanishes in the presence of 2 i^ cos. a, and we have 



L 



144 GENERAL PRINCIPLES OF MECHANICS. [§43. 

Vi = v"^ -{- 2 p V T COS. a. 

But V T is the elementary space A N = A = o, and ^ cos. a is the 
tangential acceleration, i.e., the component h of the acceleration 7; 

in the direction of the tangent or of the motion, whence we have 

2 2 

Here a cos. a is the projection,^ i^ = ^1 of the space upon the 
direction of the acceleration, and consequently we have 



Vi^ — v"" 



— V ^1. 



2 

As the motion progresses v^ changes successively into v<i,v^. , . , 
v„, and the projections of the elementary spaces are increased by 
the quantities ^.2, ^3 .... ^„, therefore we have 



Vo"" — V,'' . v."" — V^ ^ V^ — V ^ 



V ^2, ^ =i? ^3, ^ ^V ^, 



and by addition 



n— 1 ^ ^ 



vj — v^ 



;? (I'l + ^2 + ... Q =px, 



2 

in which x denotes the total projection of the acceleration upon 
A X. We can also put 



M+^2+-.. + ^. \ ^^ 



2 

when the acceleration is variable and assumes successively the val- 
ues ^1, ^2 . • • • Pw 

We see from the above that the variation of the velocity does 
not in the least depend upon the form or length of the path de- 
scribed, but only on its projection x upon the direction of the ac- 
celeration. For this reason all the jets of water, Fig. 76, have one 
and the same velocity on reaching the same horizontal plane H H. 
If c is the initial velocity or velocity of efflux, v the velocity at HH, 
and h the height of the line H H above A, we have 



v"" - & 



=z — g t^ whence 



2 

V = Vc' - %gK 
If at a certain point of the motion we have a = 90°, the tan- 
gential acceleration h ^ p cos. a becomes = 0, and the normal ac- 
celeration n = p sin. a is equal to the mean acceleration p. In this 
case the variation of the squares of the velocities while the element 
G of the space is being described, is v^^ — 2;^ = 0, and we have Vj = 
v. and if the motion continues in a curve, the direction of the ac- 



§44.] 



COMPOUND MOTION. 



145 



celeration changing in sucli a manner as always to remain normal 
to the direction of the motion (i.e., if there is no tangential accel- 
eration), 2'," — v"" = 0, or Vi = V remains constant while the point is 
describing any finite space, and the final velocity is equal to the 
initial velocity c. 

The normal acceleration, for which the velocity remains constant. 



IS 



P = 




an example of which is afforded by motion in a circle, for then the ra- 
dius of curvature C A = C = G D = r is constant. Inversely 

a constant acceleration, which always acts 
at right angles to the direction in which 
the body is moving, causes uniform mo- 
tion in a circle. 

Example, — A body, revolving in a circle 5 
feet in diameter in such a manner as to make each 

o _ ,>. 

revohition in 5 seconds, has a velocity c= —-- = 

— ^ — z= 2.7r = 6,283 feet, and a normal ac- 
5 

celeration j?=: ' ^ = 7,896 feet, i.e., in every 

second it would be diverted from the straight line a distance \ p=^ . 7,896 
= 3,948 feet. 

(§ 44.) Curvilinear Motion in G-eneral. — If a point P, Fig. 
79, moves in two directions A X and A Y Sit the same time, we 

can consider the spaces de- 
scribed A K = L F = X 
and A L = KP = y sls the 
co-ordinates of the curve 
A P T^^ formed by the path, 
and if 6? ^ is the element 
of time, in which the body 
describes the elementary 
spaces P R = d X and E Q 
= di/yWe have (from § 20) 
the velocity along the ab- 
scissa 

^> ** = ?? 

and that along the ordi- 
nate 



Fig. 79. 




146 



GENERAL PRINCIPLES OF MECHANICS 



[§44. 



and therefore the resulting tangential velocity, or that along the 
curve, when the directions A X and A Y of the motions are at 
right angles to each other, 

in which formula d s denotes the element P Q of the curve which, 
according to Art. 32 of the Introduction to the Calculus, is equal to 

Vd x" + d y". 
The acceleration along the abscissa is, according to § 20, 

,. du 

^) ^ = j? 

and that along the ordi- 
nate 

^, dv 

^) ? = ^f 

For the tangential an- 
gle P T X= Q P R = a, 
formed by the direction of 

motion P w with the direc- 
tion of the abscissas, we 
have, 

V dy 
^ u dx 



Fig 80. 










<^' and also 


sin. 


a = 


V 

' w 


= -^ and 

d s 


COS. 


a = 


u 
w 


_ dx 
~ d s' 



The accelerations p and q can be decomposed into the following 
components in the directions of the tangent P T and of the nor- 
mal P iV, 

2)^ — jj COS. a and p)^ = p sin. a, 

qi = g sin. a and q^ = q cos. a. 
Consequently the tangential acceleration is 

Jc = pi + qi = p COS. a + q sin. a 



dv 



_du u 

d t w ' d t 
and the normal acceleration is 






V 

w 



u d u + V d V 
w dt ' 



§44.] COMPOUND MOTION. 147 

n ^ p^ ^ q^^ — p sin. a — q cos. a 
_ du V dv u_vdu — udv 
d t IV d t 10 ~ 10 dt 
But by differentiating iC' -^ v' — iv^ we obtain 

th d i^ + V d V =^ lu d w, 
and therefore we have more simply for the tangential acceleration 
. -, _w d ID __d w 
~~ w d t ~ dt' 

-r, , V t , ' T J u d V — V d u 
J rom ta7ig. a = - we obtain d tang, a = -, 

u It 

(Introduction to the Calculus, Art. 8) and the radius of the curva- 
ture C P = C Q of the elementary arc P Q (according to Art. 33 
of the Introduction to the Calculus) is 

_ _ ds ' 

d x" d tang, a' 

whence it follows that 

, , „ 7, u^ds' ds^ dslds^ vf d t 

V d u—%1 d v= —10 d tana. a= — = — -=-——=: — ( -— I = — 

•^ r d x'' 2dt r \d tj r 

and that the normal acceleration is simply 

^. lif d s 10 ds w"^ 

r w d t r dt r 

Finally we have 

, , dio -. ds -J -, 

fc d s = -^r, ' d s = -r, d w =to d w; 
dt dt 

from which we obtain (as in § 20), 

8) '^-^ = fkds, 

when we suppose that while describing the space 5 the velocity 
changes from c to zo. Therefore, in curvilinear motion half the dif- 
ference of the squares of the velocities is equal to the product of the 
mean acceleration (k) and the space s. In like manner 

2) d X -\- q d y ^^ u d u + v d v ■=^ to d to, and therefore 

9) — ^r — =f{p d X + q d y) = f p d x -\- f q d y, and 

10) f Ic d s = f p d X + f q d y, or 
h d s = p d X + q d y. 

Tlie product of the tangential acceleration and the element of the 
curve is equal to the sum of the products of the accelerations along 
the co-ordinates and the corresponding elements of co-ordinates. 



148 GENERAL PRINCIPLES OF MECHANICS. [§44. 

Example. — A body moves on one axis A X with the velocity u=12t^ 
and on the other A Fwith the velocity « = 4 ^"^ — 9; required the other 
conditions of the resulting motion. The corresponding accelerations along 
the co-ordinates are 

du ^^ , dv ^ . 

and the co-ordinates, or spaces described along the axes, are 
X = f u d t = f 12 t d t = 6 f, and 



y 



= rvdt= /'(4 t''-9)dt=^t'-9t, 



in which equations the spaces count from the time t=0. The tangential 
velocity, or that along the curve, is 

w = VW + i)' = 4/144 f^T (4^- — 9y = 4/I6 t' + 72 f + 81 = 4 i'^ 4- 9, 
consequently the tangential acceleration is 

Jc = ,— = S t = the acceleration q along the ordinate. 
We have also for the space described along the curve 

8= / wdt= / {4:f + 9)dt = -^f + 9t. 

When the direction of the motion is given by the formula, 

'0 4:f — 9 I X — 9 
tana, a = - = — -— = ^ , 

4: P + 9 

we have d tang, a = d t, 

and therefore the radius of curvature of the trajectory is 

ds^ __ (4f + 9f.l2i^ (4:t^ + 9f 



r = 



dx^d tang, a 144 t' (4 f + 9) 13 ' 

Consequently the normal acceleration, which produces a constant 
change of direction of the motion of the body, is 

yi = — ' = — 12, or constant. 
The equation of the curve of the trajectory of the body is found by sub- 
stituting t = 4/ -^ in the foregoing equation, and it is 

-i/rTT-/f=(i-«)/i- 

9 

The ordinate?/ is a (negative) maximum for « = 0, i.e., for ^^=-^, or i = 
-g-, and a; = 6 . i'' = 6 . -|- = ^, and then 

^ 3*4*2 'T""""^' 

27 ^ — fit 

and on the contrary, it is = 0, for ^=' :^ --1- or « = -^ V 3, and x = -~. 

4 3 % 



; 15] COMPOUND MOTION. 149 

The curve which forms the path of the body runs at first below the axis 



= |/^it 



it cuts it at a point whose 



of abscissas, and after the time t 

81 
abscissa is « = — , and from that time it remains above the axis. 

The following table contains a collection of the corresponding values 

of i, n, «, w, X, y, tang, a, r and s, from which the curve ABODE, Fig. 81, 

is constructed. 

Fia. 81. 




1 t 

! 


u 


V 


IV 


X 


y 


tang, a 


r 


s 


( 














27 




o 


o 


-9 


9 








00 


4 





I 


12 


-s 


13 


6 


3 


5 
12 


169 
12 


31 
3 


li 


i8 





18 


27 
2 


-9 





- 27 


18 


I 

2 

i 


24 


7 


25 


24 


22 
3 


24 


12 


86 
3 


jV3 

j 


isVl 


18 


36 


81 

2 





^ 3 
3 


- 108 
675 


27 4/3 
6^ 


3 


36 


27 


45 


54 


+ 9 

148 


4 

55 


4 

1875 


364 


4 


48 


55 


75 


96 


H 

3 


48 


4 


3 ! 



§45, Relative Motion.— If two bodies are moving simal- 
taneously, a continual change in their relative positions, distances 
apart, etc., takes place, the value of which may be determined for 
any instant by the aid of what precedes. Let A, Fig. 82, be the 
point where one and B that where the other motion begins ; the first 



150 



GENERAL PRINCIPLES OF MECHANICS. 



[§46. 



Fm. 82. 










N^ 


^,.^ 


/, 




/k 


7 


./ 


^ i 


J 


X 



body passes in a given time {t) in the direction ^ X to the position 
M, and the other body in the same time in the direction B Yto 
the point JV. Now if we draw M N, this line will give us the rela- 
tive position and distance from 
each other of the bodies A and 
B at the end of this time. Draw- 
ing A parallel to M N, and 
Q "^=^— — 7^— ^r"^^ / making A = M N, the line 

A will also give the relative 
position of the bodies A and B, 
If we now draw N, we obtain 
a parallelogram, in which N is> = A M. If, finally, we make B Q 
equal and parallel to iV^ and draw Q, we obtain a new parallel- 
ogram ^ iV ft in which the one side ^ i\^ is the absolute space 
(y) described by the second body, the other side B Q is the space 
(x) described by the other body in the opposite direction, and the 
fourth corner is the relative position of the second body, that is, 
in reference to the position of the first body, which we consider 
to be fixed. Hence we can determine the relative position of a 
moving body (B) by giving to this body besides its motion (B JV) 
another, equal to but in the opposite direction from that A Mof 
the body (A), to which its position is referred, and then by com- 
bining in the ordinary way, as, e.g., by the aid of a parallelogram, 
these two motions. 

§ 46. If the motions of the bodies A and B are uniform, we 
can substitute for A M and B N the velocities c and Ci, that is the 
spaces described in one second. In this way we obtain the rela- 
tive velocity of one body when we give to it besides its own abso- 
lute velocity, that of the body to which we refer the first velocity, 
but in the opposite direction. 

The same relation holds good for 
the accelerations. If, e.g., a body 
A, Fig. 83, moves uniformly in the 
direction A C with the velocity c, 
and a body B moves in the direc- 
tion B Y, which makes an angle a 
with B Xx, with an initial velocity 
= and with the constant accele- 
ration p, we can also suppose that 
A stands still and that B possesses. 



-X 




§46.] COMPOUND MOTION. 151 

besides the acceleration p, also the velocity (— c) in the direc- 
tion B Xx parallel to A X\ the body will then describe the parabolic 
path B F. 

The spaces described in the time t in the directions B Y 

and B XiQiHQ B N = ^-^ and B M = c t, the first of which can be 

p f p f 

decomposed into the components NR = -^ cos. a and B R = ^-^r- 

sin, a, which are parallel and at right angles to A X. 

]S"ow if A C = a and C B — h are the original co-ordinates of 
the point B in reference to A, and A K ^^ x and X = y the co- 
ordinates of the same after the time t, we have, since A K = A C 
-ON- XRandXO = CB - B R, 

X =: a — c t — -;t- COS. a and y z=: o — -tt- sin. a, 

and consequently the corresponding relative velocities 
u = — c — p t COS. a and v = — p t sin. a. 
From the abscissa x we determine the time by the formula 

c 



p COS. a \p COS. a/ 



]) COS. a \p COS. a/ p) COS. a' 

and, on the contrary, from the ordinate y by the formula 

' p sin. a 
If the body B moves in the line A X towards A, we have b = 
and also a = 0, and therefore 

2) \pl p 

putting a; = 0, we obtain for the time, when two bodies will meet, 

/ _ a/^^ /^V — ^ — ^^ (^ P ^ ^"^ — ^ 
' p \p/ p p 

If, on the contrary, the body B moves in the line A X ahead 
of the body A, then a = 180°, and the distance of the former from 

p f 
the latter body \&x = a — ct + -^-, and, inversely, the time, at 

the end of which the bodies are at a distance x from each other, is 

' J) \p' p 

The corresponding velocity ^^ — — c + ;^ Hs = 0, and the dis- 

" c & 

tance a; is a minimum for t = - , and its value is x = a — ^r— 

p 2;; 



152 



GENERAL PRINCIPLES OF MECHANICS. 



[§46. 



For every other value of x we have two values for the time, one 



of which is greater and the other less than 



P 



Remark. — The foregoing theory of relative motion is often applied, not 
only in celestial mechanics, but also in the mechanics of machines. ' Let us 
consider the following case. 

A body A, Fig. 84, moves in the direction A X with the velocity c^. and 
should be met by another body 5 which has the velocity Cg ; what direction 
must we give the latter ? If we draw A B, lay off from B, c^ in the op- 
posite direction and complete with c^ and c^ a parallelogram B c^ c Cg, 
whose diagonal c coincides with A B, we obtain in the direction B c^ = Cg 
of its side, not only the direction B Yin which the body jSmust move, 

but also in the point of intersection G of the two 
directions A X and B F, the point where the two 
bodies will meet. If a is the angle B A X formed 
by A X, and i3 the angle A B Y formed by B Y with 
A B, we have 

sin. /? c^ 



Fig. 84. 




sm. a 



^'2 



This formula is applicable to the aberration of the 
light of the stars which is caused by the compo- 
sition of the velocity c^ of the earth A around the 
sun with the velocity c^ of the light of the star B. 
Here c^ is about 19 miles, and Cg about 192,000 
miles, consequently 
c. . 19 sin. a 



sm. (3 = ^^ sm. a = 



192000 



szn. a 
10105' 



Fig. 85. 



hence the aberration or the angle A B G = J, formed by the apparent di- 
rection A B of the star (which can be supposed to be infinitely distant) with 
the true direction B G or A I), is /3=:20" sin. a; and for a=90°, that is, for a 
star, which is vertically above the path of the earth (in the so-called pole of 
the ecliptic), we have jS = 20"". In consequence of this divergence we al- 
ways see a star 20" in the direction of the 
motion of the earth behind its true posi- 
tion, and consequently a star in the 
neighborhood of the pole of the ecliptic 
describes apparently in the course of a 
year a small circle of 20'' radius around 
its true position. For stars in the plane 
of the earth's path this apparent motion 
takes place in a straight line, and for 
the other stars in an apparent ellipse. 

Example. — A locomotive moves from 
A upon the railroad track A X, Fig. 85, 




§46.] COMPOUND MOTION. I53 

with 35 feet velocity, and another at the same time from B with 20 feet 
velocity upon the track B 7, which forms an angle B D X= 56° with the 
first. Now if the initial distances are A G = 30000 feet, and G B= 24000 
ieet, how great is the distance A after a quarter of an hour ? From the 
absolute velocity B E= c^ =20 feet of the second train, the inverse velo- 
city B F = c—Z^ feet of the first, and the included angle E B F= a=z 
180° - BD G= 180° — 56° = 124°, we obtain the relative velocity of the 
second train 



B a= Vc" + c/- -^ 2cc^cos.a= VZ5' + 20'^ — 2 . 35 . 20 . cos. 56° 

= 4/1225 + 400"^ 1400 co«."56"° = 4^1625 — 782^ = 4^842^ = 29,02 feet. 
For the angle G B F = (h, included between the direction of the rela- 
tive motion and the direction of the first motion, we have 

c. sin. 56° 20.0,8290 , . ^ «.^^^ ^ , . 0^0 ^a/ 

8in. cp = -^9 Qg— = -- 29~02~" ' ^^ **''• •^=^'^5690-1, whence ^=:34°,50'. 

The relative space described in 15 minutes =900 seconds is B 0=29,02 . 

900 = 26118 feet, the distance A B is= V(30000)^ + (24000)' = 38419 

24000 
30000' 



feet, the value of the angle B A G = A B F^ whose tangent is qnnnn~ ^'^^ 



is ip = 38° 40', and therefore the angle 

^ 5 = .^ -H V ^ 34° 50' + 38° 40' = 73° 30', 
:.nd the distance apart of the two trains after 15 minutes is 

AO^ ^ATB' + B~0'' — 2AB.B0cos.AB0 

— |/38419'^ + 26118' - 2 . 38419 . 26118 cos.'lZ" 30' 
= VT588190000 = 39850 feet. 









SECOND SECTION. 

MECHANICS, OK THE PHYSICAL SCIENCE OF 
MOTION IN GENERAL. 



C H APT ER I. 

FUNDAMENTAL PRINCIPLES AND LAWS OF MECHANICS. 

§ 47. Mechanics. — Mechanics (Fr. mecanique, Ger. Mechanik) 
is the science which treats of the laws of the motion of material 
bodies. It is an application to the bodies of the exterior world of 
that part of Phoronomics or Cinematics which deals with the mo- 
tions of geometrical bodies without considering the cause. Me- 
chanics is a part of Natural Philosophy (Fr. physique generale, 
Ger. Naturlehre) or of the science of the laws, in accordance with 
which the changes in the material world take place, viz., that part 
of it, which treats of the changes in the material world arising from 
measurable motions. 

§ 48- Force. — Force (Fr. force, Ger. Kraft) is the cause of the 
motion, or of the change in the motion of material bodies. Every 
change in motion, e.g., every change of velocity, must be regarded 
as the effect of a force. For this reason we attribute to a body 
falling freely a force, which we call gravity ; for the velocity of the 
body changes continually. But, on the other hand, we cannot 
infer from the fact that a body is at rest or moving uniformly that 
it is free from the action of any force ; for forces may balance 
each other without causing any visible effect. Gravity, which 
causes a body to fall, acts as strongly upon it when it lies upon a 
table, but its effect is here destroyed by the resistance of the table 
or other support. 



§49,50.] PRINCIPLES AND LAWS OF MECHANICS. I55 

§ 49. Equilibrium. — A body is in equilibrium (Fr. 6quilibre, 
Ger. Gleichgewicht), or the forces acting on a body hold each other 
in equilibrium, or balance each other, when they counterbalance 
or neutralize each other without leaving any resulting action, or 
without causing any motion or change of motion, e.g. When a 
body is suspended by a string, gravity is in equilibrium with the 
cohesion of the string. The equilibrium of several forces is de- 
stroyed and motion produced when one of the forces is removed or 
neutralized in any way. Thus a steel spring, which is bent by a 
weight, begins to move as soon as the weight is removed, for then 
the force of the spring, which is called its elasticity, comes into 
action. 

Statics (Fr. statique, Ger. Statik) is that part of mechanics which 
treats of the laws of equilibrium. Dynamics (Fr. dynamique, Ger. 
Dynamik), on the contrary, treats of forces as producers of motion. 

§ 50. Classification of the Forces. — According to their 
action, we can divide forces into motive forces (Fr. forces motrices 
puissance, Ger. bewegende Krafte), and resistances (Fr. resistances, 
Ger. Widerstande). The former produce, or can produce, motion, 
the latter can only prevent or diminish it. Gravity, the elasticity of 
a steel spring, etc., belong to the moving forces, friction, resistance 
of bodies, etc., to the resistances ; for although they can hinder or 
diminish motion or neutralize moving forces, they are • in no way 
capable of producing motion. The moving forces are either accel- 
erating (Fr. acceleratrices, Ger. beschleunigende) or retarding (Fr. 
retardatrices, Ger. verzogernde). The former cause a positive, the 
latter a negative, acceleration, producing in the first case an accel- 
erated, and in the second a retarded motion. The resistances are 
always retarding forces, but all retarding forces are not necessarily 
resistances. When a body is projected vertically upward, gravity 
acts as a retarding force, but gravity is not on this account a re- 
sistance, for when the body falls it becomes an accelerating force. 
We distinguish also uniform (Fr. constantes, Ger. bestandige, con- 
stante) and variable forces (Fr. variable, Ger. veranderliche). While 
uniform forces act always in the same way, and therefore in the 
equal instants of time produce the same effect, i.e., the same in- 
crease or decrease of velocity, the effects of variable forces are 
different at different times ; hence the former forces produce uni- 
formly variable motions, and the latter variably accelerated or 
retarded motions. 



156 GENERAL PRINCIPLES OF MECHANICS. [§51,52,53. 

• 

§ 51. Pressure. — Pressure (Fr. pression, Ger. Druck), and 
traction (Fr. traction, G-er. Zug), are the first effects of force upon 
a material body. In consequence of the action of a force bodies are 
either compressed or extended, or, in general, a change of form is 
caused. 

The pressure or traction, produced by gravity acting vertically 
downwards and to which the support of a heavy body or the string, 
to which it is suspended, is subjected, is called the weight (Fr. poids, 
Ger. Gewicht) of the body. 

Pressure and traction, and also weight, are quantities of a pe- 
culiar kind, and can be compared only with themselves ; but since 
they are effects of force they may be employed as measures of the 
latter. 

The most simple and therefore the most common way of 
measuring forces is by means of weights. 

§ 52. Equality of Forces. — Two weights, two pressures, two 
tractions, or the two forces corresponding to them are equal, when 
we can replace one by the other without producing a different 
action. When, e.g., a steel spring is bent in exactly the same man- 
ner by a weight G suspended to it as by another weight Gx hung 
upon it in exactly the same manner, the two weights, and therefore 
the forces of gravity of the two bodies are equal. If in the 
same way a loaded scale (Fr. balance, Ger. Waage) is made to bal- 
ance as well by the weight G as by another Gi, with which we have 
replaced G^ then these weights are equal, although the arms of the 
balance may be unequal, and the other weight be greater or less. 

A pressure or weight (force) is 2, 3, 4, etc., or in general n 
times as great as another pressure, etc., when it produces the same 
effect as 2, 3, 4 .... ^ pressures of the second kind acting together. 
If a scale loaded in any arbitrary manner is caused to balance by 
the weight ( G) as well as by 2, 3, 4, etc., equal weights ( G^), then is 
the weight {G) 2, 3, 4, etc. times as great as the weight {G^). 

§ 53. Matter. — Matter (Fr. Matiere, Ger. Materie) is that, by 
which the bodies of the exterior world (which in contradistinction 
to geometrical bodies are called material bodies) act upon our 
senses. Mass (Fr. masse, Ger. Masse) is the quantity of matter 
which makes up a body. 

Bodies of equal volume (Fr. volume, Ger. Volumen) or of equal 
geometrical contents generally have different weights. Therefore 



§54,55.] PRINCIPLES AND LAWS OF MECHANICS. I57 

we can not determine from the volume of a body its weight ; it is 
necessary for that purpose to know the weight of the unit of 
volume, E.G., of a cubic foot, cubic meter, etc. 

§ 54, Unit of Weight.^ — The measurement of weights or 
forces consists in comparing them to some given unchangeable 
weight, which is assumed as the unit. We can, it is true, choose this 
unit of weight or force arbitrarily, but practically it is advan- 
tageous to choose for this purpose the weight of a certain volume 
of some body, which is universally distributed. This volume is 
generally one of the common units of space. One of the units of 
weight is the gram, which is determined by the weight of a cubic 
centimetre of pure water at its maximum density (at a temperature 
of about 4° C). The old Prussian pound is also a unit referred 
to the weight of water. A Prussian cubic foot of distilled water 
weighs at 15° R. in vacuo 66 Prussian pounds. Now a Prussian 
foot is = 139,13 Paris lines = 0,3137946 meter; whence it follows 
that a Prussian pound = 467,711 grams. The Prussian new or 
custom-house pound weighs exactly J kilogramm. The English 
pound is determined by the weight of a cubic foot of water at a 
temperature of 39°, 1 F. The pound is equal to 453,5926 grams. 
A cubic foot of water weighs 62,435 lbs. 

§ 55. Inertia (Fr. inertie, Ger. Tragheit) is that property of 
matter, in virtue of which matter cannot move of itself nor change 
the motion, that has been imparted to it. Every material body re- 
mains at rest as long as no force is applied to it, and if it has been 
put in motion continues to move uniformly in a straight line, as 
long as it is free from the action of any force. If, therefore, 
changes in the state of motion of a material body occur, if a body 
changes the direction of its motion, or if its velocity becomes 
greater or less, this result must not be attributed to the body as a 
certain quantity of matter, but to some exterior cause, I.E., to a 
force. 

Since, whenever there is a change in the state of motion of a 
body, a force is developed, we can in this sense count' inertia as one 
of the forces. K a moving body could be removed from the influ- 
ence of all the forces which act upon it, it would move forward 
uniformly for ever; but such a uniform motion is nowhere to be 
found, since it is impossible for us to remove a body from the in- 
fluence of every force. If a mass moves upon a horizontal table 



158 GENERAL PRINCIPLES OF MECHANICS. [§56,57. 

the action of gravity is counterbalanced by tke table, and therefore 
does not act directly upon the body, but in consequence of the 
pressure of the body on the table a resistance is developed, which 
will be treated hereafter under the name of friction. This resist- 
ance continually diminishes the velocity of the moving body, and 
the body therefore assumes a uniformly retarded motion and finally 
comes to rest. The air also opposes a resistance to its motiou, and 
even if the friction of the body could be completely put aside, a 
continual decrease of velocity would be caused by the former. 
But we find that the loss of velocity becomes less and less, and that 
the motion approximates more and more to a uniform one, the more 
we diminish the number and magnitude of these resistances, and we 
can therefore conclude, that if all moving forces and resistances 
were removed, a perfectly uniform motion would ensue. 

§ 56. Measure of Forces. — The force (P) which accelerates an 
inert mass (if) is proportional to the acceleration {p) and to the 
mass {M) itself. When the masses are the same, it increases with 
the infinitely small increments of velocity produced in the infin- 
itely small spaces of time, and when the velocities are equal it in- 
creases in the same ratio as the masses themselves. In order to 
produce an m fold acceleration of the same mass, or of equal masses, 
we require an m fold force, and an 7i fold mass requires an n fold 
force to produce the same acceleration. 

Since we have not as yet adopted a measure for the masses, we 
can assume 

P = Mp, 

or that tlie force is equal to the product of mass and the accelera- 
tion, and at the same time we can substitute instead of the force 
its effect, i.e., the pressure produced by it. 

The correctness of this general law of motion can be proved by 
direct experiment, when we, E.G., drive along upon a horizontal 
table by means of bent steel springs similar or difterent movable 
masses; but the important proof lies in this, that all the results 
and rules for compound motion, deduced from the law, correspond 
exactly with our observations and with natural phenomena. 

§ 57. Mass,— All bodies at the same point on the earth fall in 
vacuo equally quickly, namely, with the constant acceleration 
g = 9,81 meter =: 32,2 feet (§ 15). If the mass of a body is = M 



ioJ 



.] PRINCIPLES AND LAWS OF MECHANICS. I59 



and the weight which measures the force of gravity = G, we have 
from the last formula 

G = Mg, 

I.E., the weight of a 'body is a product of its mass and the acceleration 
of gravity, and inversely 

M= —, 
9 

I.E., the mass of a tody is the weight of the same divided ly the accel- 
eration of gravity, or the mass is that weight which a body would 
have if the acceleration of gravity were = 1, E.G., a meter, a foot, 
etc. For that point upon or in the neighborhood of the earth or of 
any other celestial body, where the bodies fall with a velocity (at the 
end of the first second) of 1- meter instead of 9,81 meters, the mass, 
or rather the measure of the same, is given directly by the weight 
of the body. 

According as the acceleration of gravity is expressed in meters 
or feet we have for the masses 

M=^ = 0,1019 G, or 

J^ = -3!^ = 0,031 6^. 

Hence the mass of a body, whose weight is 20 pounds, is 
jlf= 0,031 X 20 = 0,62 pounds, and inversely the weight of a 
mass of 20 pounds i^ G — 32,2 x 20 = 644 pounds. 

§ 58. — If we suppose the acceleration (g) of gravity to be con- 
stant, it follows that the mass of a body is exactly proportional to its 
weight, and that, when the masses of two bodies are M and J/j and 
their weights G and G^, we have 

M G 



M, ~ g; 

Therefore, the weight of a body can be employed as a measure 
of its mass, so that the greater the mass a body is the greater is its 
weight. 

However the acceleration of gravity is variable, becoming 
greater as we approach the poles and diminishing as we approach 
the equator ; it is a maximum at the poles and a minimum at the 
equator. It also decreases when a body is elevated above the level 



160 GENERAL PRINCIPLES OF MECHANICS. [§59. 

of the sea. Now since a mass, so long as we take nothing from it 
nor add anything to it, is a constant quantity, and remains the 
same for all points on the earth, and even on the moon, it follows 
that the weight of a body must be variable and depend upon the 
position of the body, and that in general it must be proportional 

to the acceleration of gravity, or that -^ must be = — . 

The same steel spring would therefore be differently deflected 
by the same weight at different points on the earth— at the 
equator and on high mountains the least, and at the poles at the 
level of the sea the most. 

§ 59. Heaviness (Fr. densite, G-er. Dichtigkeit) is the in- 
tensity with which matter fills space. The heavier a body is, the 
more matter is contained in the space it occupies. The natural 
measure of the heaviness is that quantity of matter (the mass) 
which fills the unity of volume; but since matter can only be 
measured by weight, the weight of a unit of volume, e.g., of a 
cubic meter or of a cubic foot of another matter, must be employed 
as the measure of its heaviness. Hence, the heaviness of water 
at 39°.l F. is = 62,425 pounds, and that of cast iron at 32° F. 
is = 452 pounds, i.e., a cubic foot of water weighs 62,425, and 
a cubic foot of cast iron 452. In ordinary calculations we assume 
that of water to be 62 i pounds. From the volume F of a body 
and its heaviness y we have its weight G = Vy. 

The product of the volume and the heaviness is the weight. 

The heaviness of a body is uniform (Fr. homogene, uniforme, 
Ger. gleichformig) or variable, (Fr. variable, heterogene, Ger. 
ungleichformig), according as equal portions of the volume 
have equal or different weights, E.G., the heaviness of the simple 
metals is uniform, since equal parts of them, however small, weigh 
the same. Granite, on the contrary, is a body of variable heaviness, 
since it is composed of parts of different density. 

Example. — 1. If the heaviness of lead is 712 pounds, then 3,2 cubic feet 
of lead weigh Gr = Vy = 2278,4 lbs. If the weight of a cubic foot of bar 
iron be 480 pounds, the volume of a piece, whose weight is 205 pounds, is 

V =— = T^ = 0,4271 cubic feet = 0,4271 x 1728 = 738 cubic inches. 

7 480 ' 



Note. — In German and French the word " density" is employed to express 
the weight of a cubic foot, a cubic meter, etc., of any material. In English, 
unfortunately, it is employed as a synonym of specific gravity. — Tr. 



§60.] PRINCIPLES AND LAWS OF MECHANICS. 161 

If 10,4 cubic feet of hemlock, thoroughly saturated with water, weighs 
577, then its heaviness is 

Or Oil »<wp» -1 

y = '^= ^^ = 55,5 pounds. • 

§ 60. Specific Gravity. — Specific weight, or specific gravity, 
(Fr. poids specifique, Ger. specifisches or eigenthiimliches Gewicht) 
is the ratio of the heaviness of one body to that of another body, 
generally water, which is assun^ed as the unit. But the heaviness 
is equal to the weight of the unit of volume ; therefore the specific 
gravity is also the ratio of the weight of one . body to that of 
another, E.G., water, of equal volume. 

In order to distinguish the specific gravity or specific weight 
from the weight of a body of a given volume, the latter is called the 
absolute iveiglit (Fr. poids absolu, Ger. absolutes Gewicht). 

If y is the heaviness of the matter (water), to which the others 
are referred, and y^ the heaviness of any matter whose specific 
gravity is denoted by e, we have the following formula : 

e = ~ and yi = e y, 

r 

therefore the heaviness of any matter is equal to the specific gravity 
of the same multiplied by the heaviness of water. 

The absolute weight 6^ of a mass of whose volume is V, and 
whose specific gravity is e, is : 

a=Vy,= Vey. 

Example. — 1. The heaviness of pure silver is 655 pounds, and that of 

water 62,435 pounds ; consequently the specific gravity of the former (in 

655 
relation to water) is = . = 10,50, i.e., every mass of silver is 10|- 

times as heavy as a mass of water that occupies the same space. 3) If we 
take 13,598 for the specific gravity of mercury, and the heaviness of water as 
63,425, then we have for the heaviness of mercury, 

7 = 13,598. 63,435 =.- 848,86 pounds. 
A mass of 35 cubic inches of the same weighs, since 1,738 cubic inches are 
a cubic foot, 

G = 848,86 V = ?^^f|g^ = 17,19 pounds. 

Remakk. — The use of the French weights and measures possesses the 
advantage that we can perform the multiplication by e and 7 by simply 
changing the position of the decimal point, for a cubic centimeter weighs 
a gram, and a cubic meter a million grams, or 1,000 kilograms. The 
heaviness of mercury is therefore, when we employ the French measure, 
7^ = 13,598 . 1000 = 13598 kilograms; that is, a cubic meter of mercury 
weighs 13598 kilograms. 



162 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 61, 63. 



§ 61. The following table contains the specific gravities of those 
substances, which are met with the oftenest in practical mechanics. 
A complete table of specific gravities is to be found in the 
Ingenieur, page 310. 



Mean specific gravity of 
the wood of deciduous 

trees, dry = 

saturated with water = 
Mean specific gravity of 
the wood of evergreen 

trees, dry = 

saturated with water = 

Mercury = 

Lead = 

Copper, cast and dense . = 
" hammered . . . = 

Brass = 

Iron, cast, white . . . . = 
" " grey . . . . = 
" " medium . . . = 

" in rods = 

Zinc, cast = 

" rolled = 

Granite . . . . = 2,50 to 
Gneiss . . . . = 2,39 to 
Limestone . . . = 2,40 to 



0,659 
1,110 



0,453 

0,839* 

13,56 

11,33 

8,75 
8,97 
8,55 
7,50 
7,10 
7,06 
7,60 
7.05 
7,54 
3,05 
2,71 
2,86 



Sandstone . . . = 1,90 to 2,70 

Brick = 1,40 to 2,22 

Masonry with mortar made 

of lime and quarry stone : 

Fresh = 2,46 

Dry = 2,40 

Masonry with mortar made 

of lime and sandstone : 

Fresh = 2,12 

Dry = 2,05 

Brickwork with mortar 

made of lime : 

Fresh = 1,55 to 1,70 

Dry = 1,47 to 1,59 

Earth, clayey, stamped : 
Fresh ... . . . . = 2,06 

Dry = 1,93 

Garden earth : 

Fresh = 2,05 

Dry ........ = 1,63 

Dry poor earth . . . . = 1,34 



§ 62. State of Aggregation, — Bodies present themselves to 
us in three difierent states, depending upon the manner in which 
their parts are held together. This is called their state of aggrega- 
tion. They are either solid (Fr. solides, Ger. fest) or fltiid (Fr. 
fluides, Ger. fliissig), and the latter are either liquid (Fr. liquides, 
Ger. tropf bar fliissig) or gaseous ( (Fr. gazeux, aeriformes, Ger. elas- 
tisch fliissig). Solid bodies are those, whose parts are held together 
so firmly, that a certain force is necessary to change their forms or 
to produce a separation of their parts. Fluids are bodies, the 
position of Avhose parts in reference to each other is changed by the 
smallest force. Elastic fluids, the representative of which is the 
air, are distinguished from liquids, the representative of which is 



* See tlie absorption of water hy wood, polytechnische Mittheihngen, 
Vol. II, 1845. 



§ 63, 64] PRINCIPLES AND LAWS OF MECHANICS. 163 

water, by the fact that they tend continually to expand more and 
more, which tendency is not possessed by water, etc. 

While every solid body possesses a peculiar form of its own and 
a definite volume, liquids have only a determined volume, but no 
peculiar form. Gases or aeriform fluids possess neither one nor the 
other. 

§ 63. Classification of the Forces. — Forces are very differ- 
ent in their nature ; we give here only the most important ones : 

1) Gravity, by virtue of which all bodies tend to approach the 

centre of the earth. 

2) Tlie Force of Inertia, which manifests itself when a change 

in the velocity or in the direction of the moving body 
takes place. 

3) The Muscular Force of living beings, or the force produced 

by means of the muscles of men and animals. 

4) The Elastic Force, or that of springs, which bodies exhibit 

when a change of form or of volume occurs. 

5) The Force of Heat, by virtue of which bodies expand and 

contract, when a change of temperature takes place. 

6) The Force of Cohesion, or the force by which the parts of a 

body hold together, and with which they resist separa- 
tion. 

7) The Force of Adhesion, or the force with which bodies 

brought into close contact attract each other. 

8) The Magnetic Force, or the attractive and repulsive force of 

the magnet. 

Then we have the electric and the electro-magnetic forces, ^i(i. 

The resistances due to friction, rigidity, resistance of bodies, 
■etc., are due principally to the force of cohesion, which, like the 
elasticity, etc., is due to the so-called molecular force, or the force 
with which the molecules, or the smallest parts of a body, act upon 
one another. 

§ 64. Forces, how Determined. — For every force, we must 
distinguish : 

1) The point of application (Fr. point d'application ; Ger. An- 

griffspunkt), the point of the body to which the force is 
directly applied. 

2) The direction of the force (Fr. direction, Ger. Eichtung), the 

right line, in which a force moves the point of applica- 



164 GENERAL PRINCIPLES OF MECHANICS. [§65,66. 

tion, or tends to move it or hinder its motion. The direc- 
tion of a force has, Hke every direction of motion, two 
senses. It can take place from left to right, or from right 
to left, from ahove downwards, or from below upwards. 
One is considered as positive, and the other as negative. 
As we read and write from left to right, and from ahove 
downwards, it is natural to consider these motions as 
positive, and the opposite motions as negative. 
3) The absolute magnitude or intensity (Fr. grandeur absolue, 
intensite, Ger. absolute Grosse) of the force, which we 
have seen is measured by weights, e.g. pounds, kilograms, 
etc. 
Forces are graphically represented by straight lines, whose 
direction and length indicate the direction and magnitude of the 
forces, and one of whose extremities can be considered as the point 
of application of the forces. 

§ 65. Action and Reaction. — The first effect produced by a 
force upon a body is an extension or compression, combined with 
a change of form or of volume, which commences at the point of 
application, and from there gradually spreads itself farther and 
farther into the body. By this inward change in the body the 
elasticity inherent in it comes into action and sets itself in equi- 
librium with the force, and is, therefore, equal to it, but acts in the 
opposite direction. Hence, action and reaction are equal and oppo- 
site. This law is true, not only for the effects of forces acting 
by contact, but also for those acting by attraction and repulsion, 
among which the magnetic forces, and also that of gravity, must 
be counted. A bar of iron attracts a magnet exactly as much a^ it 
is attracted itself by the magnet. The force, with which the moon 
is attracted towards the earth (by gravity), is equal to the force 
with which the moon reacts upon the earth. 

The force with which a weight presses upon its support is 
returned by the latter in the opposite direction. The force, with 
which a workman pulls, pushes, etc., a machine, reacts upon the 
workman, and tends to move him in the opposite direction. When 
one body impinges upon another, the first presses upon the second 
exactly as much, as the second does upon the first. 

§ 66. Division of Mechanics. — General mechanics are di- 
vided into two principal divisions, according to the state of aggre- 
gation of the bodies : 



§ 67.] MECHANICS OF A MATERIAL POINT. 165 

1) Into the mechanics of solid or rigid bodies (Fr. mecanique 

des corps solides, Ger. Mechanik der festen oder starren 
Korper). 

2) Into the meclianics of fluids (Fr. mecanique des fluides, 

Ger. Mechanik der fliissigen Korper). The latter can 
again be divided : 

a) Into the mechanics of water and other liquids or hydraulics 

(Fr. hydraulique, Ger. Hydraulik, Hydromechanik) ; and 

b) Into the mechanics of air and other aeriform bodies (Fr. me- 

canique des fluides aeriformes, Ger. Mechanik der luft- 
formigen Korper). 

If we take into consideration the division of mechanics into 
statics and dynamics, we can again divide it into : 

1) Statics of rigid bodies. 

2) Dynamics of rigid bodies. 

3) Statics of water, etc., or hydrostatics. 

4) Dynamics of water, etc., or hydrodynamics. 

5) Statics of air (of gases and vapor) or aerostatics. 

6) Dynamics of air (of gases and vapors) or aerodynamics or 

pneumatics. 



CHAPTER II. 

MECHANICS OF A MATERIAL POINT. 

§ 67. A material point (Fr. point material, Ger. materieller 
Punkt) is a material body whose dimensions in all directions are in- 
finitely small compared with the space described by it. In order to 
simplify the discussion, we will now consider the motion and equili- 
brium of a material point alone. A (finite) body is a continuous 
union of an infinite number of material points or molecules. If 
the different points or elements of a body move in exactly the same 
manner, i.e., with same velocity in parallel straight lines, the 
theory of the motion of material point is applicable to the whole 
body ; for in this case we can suppose that equal portions of the 
mass are impelled by equal portions of the force. 



166 GENERAL PRINCIPLES OF MECHANICS. [§68,69. 

§ 68. Simple Constant Force. — Ifp is the acceleration with 
which a mass M is impelled by a force P, we have from § 56 

P 

P = M p, or inversely the acceleration p = ~. 

C 
Putting the mass M ^^ — , G denoting the weight of the body 

and g the acceleration of gravity, we obtain the force 

1) P = ^ 6^, 

g 

and the acceleration 

We find then the force (P) which moves a body with the accel- 
eration [p) by multiplying the weight {G) of the body by the 

ratio I — ) of its acceleration to that of gravity. 

Inversely we obtain the acceleration (jt?), with which a force (P) 
will move a mass Jf, by multiplying the acceleration {g) of gravity 

by the ratio \-q_ \ of the force to the weight of the body. 

Example. — Let us imagine a body placed upon a very smooth horizon- 
tal table, which opposes no resistance to its motion, but which counteracts 
the effect of gravity. If this body be subjected to the action of a horizon- 
tal force, the body yields and moves forward in the direction of th^ ibrce. 
If the weight of the body is ^ = 50 pounds and the force which acts 
uninterruptedly upon it is P = 10 pounds, it will assume a uniformly accel- 

P 10 

erated motion, the acceleration of which is « = ^7- (7 = — r 32,2 = 6,44 

G' 50 ' ' 

feet. If, on the contrary, the acceleration produced in a body weighing 

p 9 

42 pounds by a force P is ^ = 9 feet, then the force is P = — G =z 



g 32,20 

.42 = 0,031 . 378 = 11,7 pounds. 

§ 69. If the force acting upon a body is constant, a uniformly 
variable motion is the result, and it is uniformly accelerated, when 
the direction of the force coincides with the original direction of 
motion, and uniformly retarded, when the force acts in the opposite 
direction. If we substitute in the formulas of § 13 and § 14, in- 

P P 

stead of ^, its value -— = — g, we obtain the following: 



§69.] MECHANICS OF A MATERIAL POINT. 167 

I. For uniformly accelerated motion : 

P P P 

1) v = c-\-'^gt = c + 32,2 -^i^feet = c 4- 9,81 -^ t metres, 

ur (r tr 

2) s^ct ^ ^l^- = ct + lQ,l-^fiQQi = ct + 4,905 -^f metres. 

(jr 4i (r (jr 

II. For uniformly retarded motion : 

P P P 

1) ^ =: c — ~^gt — c — 32,2 -^ t feet — c — 9,81 7^ ^ metres. 

6r Cr 6r 

2) 5 = ct - — ^ = ct -16,l-p:^f feet = ct - 4:,906 ^ f metres. 

By means of the above formulas all questions, which can arise 
in reference to the rectilinear motions produced by a constant force, 
can be answered. 

Example. — 1) A wagon weighing 2,000 pounds moves upon a horizon- 
tal road, which opposes no resistance to it, with a velocity of 4 feet, and 
is impelled during 15 seconds by a constant force of twenty-five pounds ; 
with what velocity will it proceed after the action of this force ? The 

P 

required velocity is v = c + 32,2 —^ t, but here we have c = 4, P = 25, 

(x 

25 
G = 2,000 and t = 15, whence « = 4 + 32,2 . ^r^r^ . 15 = 4 + 6,037 = 

10,037 feet. 2) Under the same circumstances a wagon weighing 5,500 
pounds, which in the three previous minutes had described uniformly 950 
feet, was impelled during 30 seconds by a constant force, so that after- 
wards it described 1650 feet uniformly in three minutes. ' What was this 

force ? The initial velocity is c = -; — -r-^ = 5,277 feet, and the final ve- 

•^ 3 . 60 ' ' 

1650 P 

locity is ?j = = 9,166 feet, whence —- g t = v — c =-. 3,889, and the 

o gQQ Q. 5500 550 

force P = ' ' — ~ = 0,031 . 3,889 . --- = 0,120559 .— -= 22,10 pounds. 
g t oO o 

3) A sled weighing 1500 pounds and sliding on a horizontal support with 
a velocity of 15 feet loses, in consequence of the friction, in 25 seconds, the 
whole of its velocity. What is the amount of the friction ? The motion is 
here uniformly retarded and the final velocity is « = 0, hence c = 32,2 . 

~, and P = 0,031 ? ^ = 0,031 ""^^-^ = 0,031 . 900 = 27,9 pounds, 

which is the friction in question. 4) Another sled, weighing 1200 pounds 
and moving with an initial velocity of 12 feet, is obliged to overcome a 



16b GENERAL PRINCIPLES OF MECHANICS. [§70. 

friction of 45 pounds wlien in motion. What is its velocity after 8 seconds, 
and what is the space described ? 
The final velocity is 

t) = 12 - 32,2 y^-- = 12 - 9,66 = 2,34 feet, 

and the space described is 

Ic + 'D\ /12 + 2,34\ 

s = ( - J-) t = (— ^ ) • S = ^^^^^ ^ee*- 

§ 70. Mechanical Effect.* — Mechanical effect or icork done 
(Fr. travail mecanique, Ger. Leistung or Arbeit der Kraft) is that 
effect which a force accomplishes in overcoming a resistance, as, 
E.G., gravity, friction, inertia, etc. Work is done when we elevate 
a weight, when a greater velocity is communicated to a body, when 
the forms of bodies are changed, when they are divided, etc. The 
work done depends not only upon the force, but also on the space 
during which it is in action, or during which it overcomes a re- 
sistance. If we raise a body slowly enough to be able to disregard 
the inertia, the work done is proportional to its weight and to the 
height which it is lifted for 1) the effect is the same if a body of 
the 771 (3) fold weight is lifted a certain height, or if m (3) bodies 
of the weight {G) are lifted the same height; it is m times as 
great as that necessary to raise the simple weight the same height ; 
and in like manner 2) the work done is the same, if one and the 
same weight be raised the n (5) fold height (n h) or if it is raised 
n (5) times to the simple height, and in general n (5) times so great, 
as when it is raised . to the simple height. In like manner, the 
work done by a weight sinking slowly is proportional to the weight 
and to the distance it sinks. This proportion is, however, true for 
every other kind of work done ; in order to make a saw cut of 
twice the length and of the same depth as another we are obliged 
to separate twice as many particles, and the work done is therefore 
double ; the double length requires the force to describe double the 
distance, and consequently the work is proportional to the space 
described. In like manner the work done by a run of millstones 
increases evidently with the number of grains of a certain kind 
of corn which it grinds to a certain fineness. This quantity is, 
however, under the same circumstances proportional to the number 

^ * Energy is the capacity of a body to perform work. Energy is said to be 
stored when this capacity is increased, and to be restored when it is diminished. 
The unit of energy is the same as that of work. — Tr. 



§ 71, 72.] MECHANICS OF A MATERIAL POINT. 169 

of revolutions, or rather to the space described by the upper mill- 
stone while this quantity of corn is being ground. The work 
done increases, therefore, directly with the space described. 

§ 71. As the work done by a force depends upon the inten- 
sity of the force and the space described by it, we can assume as 
the unit of luorh or dynamical unit (Fr, unite dynamique, Grer. 
Einheit der mechanischen Arbeit oder Leistung) the work done 
in overcoming a resistance, whose intensity is the unit of weight 
(pound, kilogram) over a space equal to the unit of length (foot, 
metre), and we can also put this measure equal to the product 
of the force or resistance into the space described by it in its 
direction while overcoming the resistance. 

If we put the amount of the resistance itself = P and the 
space described by the force, or rather by its point of application, 
while overcoming it = s, then the work done in overcoming this 
resistance is 

A = P s units of work. 

In order better to define the units of work (which we can style 
simply dynam) the units of both factors P and s are generally 
given, and instead of units of work we say kilogram-meters and 
pound-feet, or inversely meterkilograms, foot-pounds, etc., accord- 
ing as the weight and the space are expressed in kilograms and 
meters, or in pounds and feet. For simplicity we write instead of 
meterkilogram, m Ic ov k m; and instead of foot pound, lb. ft., 
or ft. lb. 

Example. — 1. In order to raise a stamp weighing 210 pounds, 15 inches 

15 
high, the work to be done is yl = 210 . zr^ = 262,5 ft. lbs. 2. By a me- 

cbanical effect of 1500 foot pounds a sled, which when moving must over- 
come a friction of 75 pounds, will be drawn forward a distance 

A 1500 -^. , 
« = p = -if^ = 30 feet. 

§ 72. Not only when the force is invariable, or the resistance is 
constant, but also when the resistance varies while the force is 
overcoming it, can the work done be expressed by the product of the 
force and the space described, provided we assume for the value of 
the force the mean value of the continuous succession of forces. The 
relation between the time, velocity and space is therefore the same 
here ; for we can regard the latter as the product of the time and 



170 



GENERAL PRINCIPLES OF MECHANICS. 



[§73. 



of the mean of the velocities. Y\^e can also employ here the same 
graphical representations. The work done can be regarded as the 
area of a rectangle A B G D, Fig. 86, whose base ^ ^ is the space 
(s) described and whose height is either the constant force P or 
the mean value of the different forces. In general, however, the 
work done can be represented by the area of a figure A B C N D, 
Fig. 87, the base of which is the space s described, and the height 



D 



Fig. 86. 

N 



M 



B 




of which above each point of the base is equal to the force corre- 
sponding to that point of the path. If we transform the figure 
A B C N D in-di rectangle A B E Fy^ilh the same base and the 
same area, its altitude A F = B E gives the mean value of the 
force. 

§ 73. Arithmetic and Geometry give several different methods 
for finding the mean value of a continuous succession of quanti- 
ties, the most important of which are to be found in the Ingcmieur, 
The method known as Simpson^s Rule is, however, the one most 
generally employed in practice, because in many cases it unites 

great simplicity with a high degree of 
accuracy. 

In every case it is necessary to divide 
the space A B = s (Fig. 88), in n (as 
many as possible) equal parts, such as 
A E = E G = GJ, etc., and to deter- 
mine the forces EF= P^, G H = P., J K 
= Pz, etc., at the ends of these portions 
of the path. If we put the initial force 
A D = P^ and the final one B G — P,, 
we have the mean force P = (^ P^ + ^i + ^2 + ^3 + • • • + 
P„_i + A P„) : n, and consequently its work 



Fig. 88. 




P 5 = (i P, + P, + P, + . . . + P„„: + i PJ 



n 



§ 74.] MECHANICS OF A MATERIAL POINT. 171 

If the number of parts {71) be even, i.e. . 2, 4, 6, 8, etc., Simp' 
son's Enle gives more exactly the mean force 

P = (P„ + 4 P, + 2 P., + 4P3 + . . . +4P„_i + P„) : dn, 
whence the corresponding work done is 

Ps = (P„ + 4 P, + 2 P, + 4 P3 + .. . + 4 P„_, 4- P„) ^. 

on 

If n is an uneven number, we can put 
Ps= [i{F, + dP, + 3P, + P,) + i (P3 + 4 P, 4- 2 P5 

+ .... + 4 P„_i + P„)] — . (See Art. 38 of the Introduction 

n 

to the Calcukis.) 

Exajviple, — In order to determine the work done by a horse, in drawing 
a wagon along a road, we employ a dynamometer (or force measurer), one 
side of which is attached to the wagon and the other to the horse, and we 
observe from time to time the intensity of the force. If the initial force is 
P = 110 pounds, that after movmg 25 feet 122 pounds, that after 50 feet 127 
pounds, that after 75 feet 120 iDOunds, and that at the end of the whole dis- 
tance, 100 feet, 114 pounds, we have for the mean value of the force ac- 
cording to the first formula 

p = (^ . 110 + 122 + 127 4- 120 + I X 114) : 4 = 120,25 pounds, 
and for the mechanical effect 

Ps = 120,25 X 100 = 12025 foot-pounds. 

According to the second formula we have 

1446 
P=(110 + 4. 122 + 2. 127 + 4. 120 + 114) : (3 . 4) = ——==120,5 pounds, 

and the mechanical effect 

P« = 120,5 . 100 = 12050 foot-pounds. 



■^i 



74. Principle of the Vis Viva or Living Forces. — If in 



V — c"" v"" — c 



the formula s = —^ or p s = — ^ — , found in 8 14, we substi- 

2j!? 2 

P 

tute for p its value -^ g, we obtain the mechanical effect A = P s 

(y^ (f\ y" 

— _ jG, or designating the heights due to the velocities—- 



and X-- by h and h, 



P s = Qi-lc) G, 



This equation, so important in practical mechanics, means that 
the mechanical effect (P 5), which a mass absorbs when its velocity 
changes from a lesser to a greater, or that which it gives out, when 
its velocity is forced to change from a greater to a less, is always 



172 GENERAL PRINCIPLES OF MECHANICS. [§75. 

equal to the product of the weight of the mass into the difference of 
the heights due to the different velocities (^— — k— )• 

Example. — 1. In order to impart, upon a perfectly smooth railroad, a 
velocity of 30 feet to a wagon weighing 4000 pounds, the work to be done 

is Ps =~ G= 0,0155 v"" G = 0,0155 x 900 x 4000 = 55800 pounds, and this 
2g 

wagon will perform the same amount of work if a resistance be opposed to 

it, so as to cause it gradually to come to rest. 2. Another wagon, weighing 

6000 pounds and moving with a velocity of 15 feet, acquires in consequence 

of the action of a force a velocity of 24 feet ; how much mechanical effect 

is stored by the wagon, or how much work is performed by the force ? 

The heights due to the velocities 15 and 24 feet are ^ = ^r— = 3,487 and 

2g ' . 

^ = -— = 8,928 feet. Consequently the work done P s = {h — k) G 

= (8,928 - 8,487) x 6000 = 5,441 x 6000 = 32646 foot-pounds. 

If the space described is known the force can be found, and if the 
force is known the space can be found. Let us suppose, e.g., in the last 
case, that the space described by the wagon, while the velocity changes from 
11 to 24 feet, is but 100 feet, we have then the force P = (h — k) 

— = -^TCTj — = 326,46 pounds. If, however, the force was 2000 pounds, 

the space would be s = (^ — ^) ^ = - = 16,323 feet. 3. If a sled 

weighing 500 pounds, and moving with a velocity of 16 feet, loses in con- 
sequence of the friction the whole of its velocity while describing 100 feet, 
the resistance of the friction is 

P = '^-— ? = 0,0155 X 16^^ X ^ = 0,0155 x 256 x 5 = 19,84 pounds. 
8 ' 100 ^ 

§ 75. The formula for the work done, found in the preceding 
paragraph, 

P ^ = (^^i~) <^ = ('^ - ^) Cf' 

holds good not only when the forces are constant, but also when 
they are variable, if we substitute (according to § 73) instead of F 
the mean value of the force; for according to III*), in § 19, we 
have, in general, for every continuous motion 



v' - e 



2 



= 2> s, 



in which p = '----- — -- denotes the mean acceleration 

71 



s and 



§75.] MECHANICS OF A MATERIAL POINT. 173 

with which the space s is described, and we have also 

p = =rp , whence 

J" . = (-^-) Jf = '-^^ ff = (S - I) 9. 

in which P = — '-^ denotes the mean of all the forces 

n 

measured after the spaces — , — , — . . . — are described. 

n n n n 

The force F can also be calculated by means of one of the 
formulas of § 73, when the number n of the parts is not assumed 
to be very great. 

We are very often required to calculate the change of velocity 
that a given mass M undergoes, when a given amount of me- 
chanical effect P s is imparted to it. The principal equation 
which we have found is then to be employed in the following form 

h — h -\ — 7^ OT V — y c -^ 'Zg -7^. 

If we have calculated by means of this formula the velocities 

Vi, Vi . , , Vn which correspond to the spaces — , — , — ... s, we can 

calculate by means of the formula 

5/111 1\ 

n \vi V2 Vs vj 

the time in which the space s is described. 

2 P 5 P s 

In the form 6^= if ^ = -j — -^ = y-; -r-? \ ^^^ principal 

V — C Tf iV ~x~ C) [V — C) 

formula we have found serves to determine the mass M, which in 
consequence of the mechanical effect P s imparted to it will un- 
dergo a change of velocity v — c. 

When the motion of a body is continuous and the final velocity 
V is equal to the initial one c, then the work done becomes = 0, 
I.E, the accelerated part of the motion absorbs exactly as much 
work as the retarding portion gives out. 

Example. — If a wagon weighing 2500 pounds, moving without fric- 
tion with an initial velocity of 10 feet, has imparted to it a mechanical 
effect of 8000 foot-pounds, what is its final velocity ? 



174 GENERAL PRINCIPLES OF MECHANICS. [§76. 

Here v = y 10^ + 64,4 . |^^ = VlOO "+"206;08 = 17,49 feet, 

Remark. — We call, without attaching any particular idea to the term, 

C 
the product of the mass M= — into the square of the velocity (v"^), that is 

M'v'^, the vis viva (Fr. force vive, Ger. lebendige Kraft) of the moving mass, 
and we can therefore put the mechanical effect, which a mass which is 
moved absorbs, equal to the half of its vis viva. If an inert mass passes 
from a velocity c to another v, then the work gained or lost is equal to the 
half difference of the vis viva at the beginning and of that at the end of 
the change of velocity. This law of the mechanical effect bodies produce 
by virtue of their inertia is called the principle of vis viva (Fr, principe 
des forces vives, Ger. Princip der lebendigen Krafte). 

§ 76. Composition of Forces. — If two forces P] and Pg act 
upon the same body 1) in the same or 2) in opposite direc- 
tions, then their effect is the same as when a single force equal to 
1) the sum or 2) the difference of these forces acted upon the body;, 
for these forces impart to the mass the accelerations 

Pi . P. 

^i = ^and^, = ^; 

whence, according to § 28, the resulting acceleration is 

p = p,±p,^ — — , 

and consequently the corresponding force is 

P :=Mp = P,± P,. 
We call the force P derived from the two forces and capable of 
producing the same effect (equipollent) their resultant (Fr. result- 
ante, Ger. Resultirende), and its constituents Pi and Ps its com- 
ponents (Fr. composantes, Ger. Componenten). 

Example. — 1) A body lying upon the flat of the hand presses with its 

absolute weight on it only so long as the hand is at rest, or is moved with 

the body uniformly up or down ; but if we lift the hand with an accelerated 

motion, it experiences a heavier pressure ; and if, on the contrary, we allovv' 

it to sink with an accelerated motion, then the pressure becomes less than 

the weight, and even = when the hand is lowered with an acceleration 

equal to that of gravity. If the pressure on the hand is P, then the body 

C 
falls with the force O — P only, if its mass is M = — ; if we put the ac- 

G 

celeration with which the hand descends = /> we have G — P = — ^, and 

therefore the pressure P = (t (?= [1 — -— I (r. If, on the contrary, 



^76.] 



MECHANICS OF A MATERIAL POINT. 



175 



we raise the body upon the hand with an acceleration ^, then the accelera ■ 
tion p is opposite to the acceleration g, and the pressure becomes P = ll 



+ 



)»■ 



According as we lower or raise a body witli an acceleration of 

20 



20 feet, the pressure upon the hand is ( 1 — -— - j G = {1 — 0,62) G = 

0,38 times the weight of the body, or 1 + 0,62 = 1,62 times the same 

weight. 2) If with the flat of the hand I throw a body weighing 3 

pounds 14 feet vertically upward, by urging it on continuously during the 

first two feet, then the work done is P s = G^ A = 3 . 14 = 42 pounds, and 

42 
the pressure of the body on the hand is P = -^ = 21 pounds. Hence 

the body when at rest presses with a weight of three pounds upon the 
hand, and, on the contrary, during the act of throwing it, it reacts with a 
force of 21 pounds upon the hand. 

3) What load Q can a piston movable in a cylinder A A CO, Fig. 89, 
raise to the height I) K = s = Q feet, if during the first half of its course 

the air which flows in from a very large res- 
ervoir acts upon it with a force of 6000 
pounds, and if during the second half of its 
course this air enclosed in the cylinder ex- 
pands according to the law of Mariotte, while 
the exterior air acts with a constant pressure, 
of 2000 pounds in the opposite direction.. 
Since the air shut in the cylinder at the end 
of the second half of the course of the piston 
has expanded to double its volume, the 
pressure of the same upon the piston at the 
end of the course is only |- . P = 3000 pounds. 
The air inclosed in the cylinder, when the 
piston has traveled 3 feet, presses with a force of 6000 pounds upon it, on 
the contrary at the end of four feet with a force of f . 6000 = 4500 pounds, 
at the end of 5 feet with f . 6000 = 3600 pounds, and at the end of the 
entire course with a force of f . 6000 = 3000 pounds. Hence the mean 
force during the expansion = ^ [6000 + 3 (4500 + 3600) + 3000] = 




33300 



8 



- = 4162 pounds, and consequently the mean force during the whole 



of the course of the piston is = 



6000 + 4162 

2 



5081 pounds. If we sub- 



tract the constant opposing force of 2000 pounds from this, it follows that 
the weight to be raised by the piston is 

Q = 5081 — 2000 = 3081 pounds. 
The motive force for the first half of the course is then P — (Q -f- 
2,000) = 6000 — 5081 = 919 pounds, and consequently the acceleration 

P-(Q + 2000)\ 919 „ ^ 

Q ) ^ = 3081 • ^''^ = ''' ^^^*' ^'"'^ 



of the motion is p = i 



176 GENERAL PRINCIPLES OF MECHANICS. [§76. 



the velocity at the end of the first half of the course of the piston s^ = -— 

— 3 feet is ?; = \I~%Y^^ = V 679,6 = V"57^ = 7,589 feet, and the time in 

which this space is described by the piston is t^ = — - = ^ ^^ = 0,790 

seconds. The distance, which has been traveled by the piston when the 

force and the load balance each other, that is, when the motive force and 

consequently the acceleration is = 0, and the velocity of the piston is a 

maximum, is 

/ P \ s 6000 .3 , , 

When the distance ~^-^r — = 3,2715 feet has been described, the force act- 

, . . 6000 . 3 

mg on the mside piston is ■■--'—-- = 5502, and consequently the motive 

o,^715 

force is = 5502 — 5081 = 421 pounds, and the mean value of the same 

while the piston passes from 3 to 3,543 feet is ~ = 434 

434 434 . 32 2 

pounds. The corresponding mean acceleration is = —^r-r fj = — 

t>081 8081 

= 4,535 feet, and consequently the maximum velocity of the piston at the 

end of the space x = s^ +33 = 3,543 feet is 

Vm = ^'v^ + 2ps~— V 57,6 + 2 X 4,535 >ro^43 = V 62,525 = 7,907 feet. 
The time required to describe the space s^ = 0,543 can be put 

= *^ = f (v + i) = »'^"^ (ri + 7^7) = "'""> ^«'=°"^^- 

If the piston has described the space 5,5 the motive force is -^-^7^ — 

0,500 

5081 = — 1808 pounds, and if the piston is midway between this point 

and the point of maximum velocity, this force is then = j — 5081 

1808 X 32 2 
= — 1100 pounds, and the corresponding accelerations are=— — ^- 

3081 

= — 18,89 feet, and = — — — '- = — 11,49 feet. 

0O8I 

The mean acceleration while the piston describes the portion of the 
space 5,500 - 3,543 = 1,957 feet is consequently = — 0+^ x 1M9 + 18,89 

= — 10,81 feet, and therefore the velocity acquired at the end of this space is 

= V62,525 - 2 X fo;81 x 1,957 = V20,215 = 4,496 feet. On the contrary, 

during the first half of the last portion of the ccurse, the mean acceleration is 

+ 11,49 



2 



= — 5,745 feet, and therefore the velocity at the end of the 



space 4,5215 feet v^ = V 62,525 — 2 x 5,745 x 0,9785 = V51,282 = 
7,161 feet, and we have for the time required to describe the space «g = 



77.] 



MECHANICS OF A MATERIAL POINT. 



177 



1'^^^' ^3 = T(i + i + i) = '^''' (7^7 + 7^1 + 4i) - ^'^^^ 

X 0,9075 = 0,396 seconds. Finally, we can put the time during which the 

2 s 1 

last portion 5^ = 0,5 of the whole course is described t^ = * — 



v^ 4,496 



= 0,2224 seconds, and the time required by the piston to describe its entire 
course t^t^+t^ + t^+ t^= 0,790 + 0,070 + 0,296 + 0,2224 = 1,378 seconds. 

]J5 § 77. Parallelogram of Forces.— If a mass (a material 
point) M, Fig. 90, is acted upon by two forces, Pi and P,, whose 
direction, M X and M F, form an angle X M Y = a with each 
other, the forces cause in these directions the accelerations 

F P 

and by combining them, a resulting acceleration (§ 35) in the 

direction M Z, which is determined by 
the diagonal of a parallelogram con- 
structed with pi, po, and a, is obtained ; 
this resulting acceleration is 

p zzz Vj)i^ + p.2 + ^PxP^ COS. a, 
and we have for the angle 0, which its 
direction makes with the direction 
M X of the acceleration pi 

»2 sin. a 

sm. (b = . 

P . 

Substituting in these two formulas the given values of pi and p 

we obtain 

Po\ sin. a 




2> 



sin.<l> = {-g) 



V 



and multiplying the first equation by M, we have 

Mp ^ V P^' + P/ 4- 2 Pi P.cos.a, 
or since Mp is the force P corresponding to the acceleration p, we 

find 1) P = V P,' + P^^ + 2 Px P2 COS. a, 

rtx . ^ P2 sin. a 
2) sm. (f) = p — 

The resultant or mean force is determined in magnitude and di- 
rection from the component forces in exactly the same manner, as the 
resulting acceleration is determined from the component accelerations. 

If we represent the forces by right lines, making the ratio of 
12 



178 



GENERAL PRINCIPLES OF MECHANICS. 



[§77. 



their length the same as that of the weights (e.g. pounds) to 
each other, the resultant can th^n be represented by the di- 
agonal of the parallelogram whose sides are formed by the compo- 
nent forces, and one angle of which is equal to the angle formed by 
the component forces with each other. The parallelogram thus 
constructed with the component forces, the diagonal of which rep- 
resents the resultant, is called the 'parallelogram of forces. 



Example. — If a body, Fig. 91, weighing 150 pounds and resting on a 
perfectly smooth table, is acted on by two forces P^ = 30 pounds, and 
Pg = 24 pounds, which form with each other an angle P^M P^ = a = 105°, 
in what direction and with what acceleration will the motion take place ? 

Since cos. o. = cos. 105° = — cos. 75°, we have 

the resultant 

P = V30' + 24^ — 2 X 24 X 30 x cos."75" 




= V900 + 576 — 1440 cos. 75" 

= V1476 -~372,^ = 33,22 pounds ; 
and the corresponding acceleration 



P=M 



P Pg 33,22 X 32,2 



150 



= 7,13 feet. 



The direction of the motion forms an angle (t> 
with the direction of the first force, which is de- 
termined by the following formula 

24 
sin. ^ = -^—^ sin. 105°=0,7224 sin. 75°=0,6978; 

and is = 44° 15'. 

Remark. — The resultant (P) depends (according to the formula just, 
found) upon the components alone, and not upon the mass ( M) of the 
body upon which the forces act. For this reason we find in many works 
upon mechanics the correctness of the parallelogram of forces demonstrated 
without reference to the mass, but with the assumption of some one of the 
fundamental laws of statics. Such pure statical demonstrations are 
numerous. In each of the following works we find a difierent one: 
"Eytelwein's Handbuch der Statik fester Korper;" " Gerstner's Hand- 
buch der Mechanik ;" " Kayser's Handbuch der Statik ;" " Mobius' Lehr- 
buch der Statik ;" " Riihlman's Technische Mechanik." The demonstration 
in Gerstner's " Mechanik" is based upon the theory of the lever ; it is really 
very simple, and is to be found in old, and also in later works, e.g., in those 
of Kastner, Monge, Whewell, etc. Kayser's demonstration is that of Poisson 
in elementary shape. Mobius' discussion of it is based upon a particular 
theory of couples (des couples) introduced by Poisson (Elements de 
Statique). A peculiar demonstration is given by Duchayla in the Corre- 
spondence sur I'ecole polytechnique No. 4, which is reproduced by Brix in 



§78.] MECHANICS OF A MATERIAL POINT. 179 

his Lehrbuch der Statik fester Korper. It is also given in many other 
works, E.G., in Moseley's Mechanical Principles, etc. The demonstration 
of the parallelogram of forces given by Navier in his " Leeons de Mecan^ 
ique" (German by Meier, 1851) is also to be found in Riihlmann's " Grund- 
ziige der Mechanik," Leipzig, 1860. A theory of this i3arallelogram, 
founded on the laws of motion, is to be found in Newton's " Principia." 
It is also employed in many later works, i.e., by Venturoli, Poncelet, Burg, 
etc. See " Elementi di Mecanica e d'ldraulica di Venturoli," " Mecanique 
industrielle par Poncelet," '' Compendium der popularen Mechanik and 
Machinenlehre von Burg." A new demonstration by Mobius is to be found 
in the Berichten der Gesellshaft der Wissenshaften zu Leipzig (1850), an- 
other by Ettingshausen in the papers of the Academy of Vienna (1851), and 
a third, by Schlomich in his " Zeitschrift fiir Mathematik and Physik" 
(1857). 

§ 78. Decomposition of Forces. — With the aid of the paral- 
lelogram of forces we can not only combine two or more forces so 
as to find a single resultant, but also decompose a given force, 
under given circumstances, into two or more forces. If the angles 
<f) and ip, which the components M P^ = Fi and MF.2 = P^, Fig. 91? 
make with the given force M P = P are given, then the compo- 
nents are determined by the following formulas 

_ P sin. ip T) _ P ^^^- ^ 
-^1 — zjT^ri \ i"\' ^2 — '777: 



sin. {(p + ipy " sin. (0 + -0)' 
If the components are at right angles, then + 1/; = 90° and sin. 
(0 + 0) r= 1, and we have 

Pi = P cos. and P,> — P sin. 0. 
and if, finally, and are equal, we have 

p _ r, __ P' Si7l. _ P 

sin. 2 2 cos. 
Example.— 1) How heavily will a table A B, Fig. 92, be pressed by a 
body Jf whose weight is G^ = 70 pounds, and which acted on by a force 

P = 50 pounds, which is inclined to the horizon 
Fig. 92. at an angle P M P^ = =z 40° ? The horizontal 

component is 

P^ = P COS. = 50 COS. 40» = 38,30 pounds, 
and the vertical component 
AgiB^;^^^^^^3 ^ = P sin. (/> = 50 sin. 40o = 32,14 pounds. 

The latter tends to raise the body from the table, 
and consequently the pressure on the table is 
G^ — Pj. = 70 — 32,14 := 37,86 pounds. 
2) If a body M, Fig. 91, weighing 110 pounds, 
is moved upon a honzontal support by two forces, 
^ so that in the first second it describes a distance 

of 6,5 feet in a direction, which forms with the two directions of the forces 



180 



GENERAL PRINCIPLES OF MECHANICS. 



[§79. 



the angles ^ = 52*^ and ip = 77^, tlie forces can be found as follows : The 
acceleration is double the space described in the first second, or in this 
case^ = 3 . 6,5 = 13 feet, and the resultant is 



P = 



p a 
g 



= 0,031 . 13 . 110 = 44,33 pounds. 



Hence one of the components is 

Psin. 770 44,33 sm. 77o 



P = 
1 sin. (530 + 770)' 



sin. 51" 



= 55,58 pounds, 



and the other is 



44,33 sin. 52o , , 

Po = ; — zttt; — = 44,95 pounds. 

^ sm. 510 ^1 



Fig. 93. 



§ 79. Composition of Forces in a Plane. — In order to find 
the resultant P of a number of component forces Pj, Pc„ P3, etc. 
(Fig. 93), we can pursue exactly the same method that we em- 
ployed in the composition of velocities. We can, by employing 
repeatedly the parallelogram of forces, combine the forces two by 
two so as to form one, until but one is left. The force Pi and Pg 
give, E.G., by means of the parallelogram M Pi Q P^, the resultant 
M Q = Q ; and if we combine this with P3 we obtain, by means 

of the parallelogram M Q R P^ 
the resultant M B = B, and the 
latter, combined with P4, gives^ 
by means of the diagonal M P 
= P, the resultant of all four 
forces Pi, P2, P3, and P4. It is 
not necessary, when combining 
these forces, to complete the par- 
allelograms and to find their 
diagonals. We have but to con- 
struct a polygon M P^ Q B P 
by drawing its sides M Pj, Pi Q,. 
Q By B P, equal and parallel to the given components Pi, P., P3, 
P4. The last side M P, which closes the parallelogram, is the re- 
sultant required, or rather the measure of the same. 

Remakk. — The solution of mechanical problems by construction is very 
usefal. Although the results are not as accurate as those obtained by cal- 
culation, yet they are of great value as checks against gross errors, and can 
therefore always be employed as proofs of calculations. In Fig. 93 we 
have drawn the forces as meeting each other and forming the given angles 
Pj MP^ = 73° 30', P^ M P^ = 33° 20', and P^M P^=^ 92° 40'; and 
their length is such, that a pound is represented by a line or -^^^ of a 




§80.] 



MECHANICS OF A MATERIAL POINT. 



181 



Prussian inch. The forces P^ = 11,5 pounds, Pg = 10,8 pounds, Pg = 
8,5 pounds, and P^ = 12,2, are therefore expressed by sides 11,5 lines, 
10,8 lines, 8,5 lines, and 12,2 lines long. A careful construction of the 
polygon of forces gives the value of the resultant P = 14,6 pounds, and 
the angle formed by the direction MP with the direction M P^ of the first 
force a = 86 f. 

§ 80. AYe can determine the resultant P more simply by de- 
composing each of the components F^, Pj? ^3? etc., into two com- 
ponents Qi and i^i, Q^ and R^, Q^ and E3, etc., in the direction of 
the rectangular axes X X and Y Y, Fig. 94, by then adding alge^ 
braically the forces which lie in the same axis, and by seeking the 
intensity and direction of the resultant of the two forces which 
have been thus obtained, and whose directions are at right angles 
to each other. If the angles Pi MX, P, MX, P3 MX, etc., P„ 
P2, Pz, etc., form with the axis of X are =■ a^, a.^, a^, etc., we 
have the components Qi = Pj cos. a^, R^ = P^ sin. a^; Q^ -— Pi 
COS. a^, R.2 — P. 2 sin. a,, etc. ; whence it follows from the equation 

e = ft + ft + ft + ..., that 

1) Q = P^ cos. Oj + P^ COS. a.2 + Pg cos. a^ + . . ., 
and also from R = R^ + P^ + R3 . . ., that 

2) R = Pi sin. a^ + P. 2 sin. a.2 -\- P3 sin. a^ + . . . 

We find the yalue of the resultant of the two components Q and R, 
just obtained, by the aid of the formula 

3) P = V Q' + R% 

and that of the angle P M X — a, formed by its direction with -the 
axis X X, by means of the formula 
R 



4) tang, a 




In adding algebraically the forces 
we must pay particular attention to 
their signs ; for if they are different 
for two different forces, i.e. if these 
forces act in opposite directions 
from the point of application, then 
this addition becomes an arithmeti- 
cal subtraction. The angle a is 
acute as long as R and Q are posi- 
tive ; it is between 90° — 180", when 
Q is negative and R positive ; it is 
between 180"— 270", when Q and R 
are both negative, and is finally be- 



tween 270" — 360", when R alone is negative. 



182 GENERAL PRINCIPLES OF MECHANICS. [§81. 

Example. — What is the direction and intensity of the resultant of 
the forces P^ = 30 pounds, P^ = 70 pounds, and Pg = 50 pounds, whose 
directions lie in the same plane and form the angles P^ ^ ^2 = 56'' and 
P^MP^ = 1040 with each other ? If we lay the axis X X, Fig. 94, in the 
direction of the first force, we obtain a^ = O^*, a^ = 56*^, and Cg = 56** + 
1040 = 1600 ; hence 

1) ^ = 30 . COS. 00 + 70 . COS. 560 + 50 cos. I6O0 = 30 + 39,14 — 46,98 
= 23,16 pounds, 

3) i? = 30 . sin. 00 + 70 . sin. 56o + 50 .sm. I6O0 = + 58,03 + 17,10 
== 75,13 pounds, and 

3) tang a = ^^j- = 3,3903, 

and therefore the angle formed by the resultant with the positive portion 

of the axis i¥Xis a = 73o 34', and the resultant itself is P =Vg' + B' = 

Q B 75,13 75,13 _ „„ 

= ~- = ~ — r,on o^/ = ttttptt; = 78,33 pounds. 

cos.a sm. a sm. 7S^ 34:' 0,9591 ' ^ 

< 

§ 81. Forces in Space. — If the direction of the forces do not 
lie in the same plane, we pass a plane through the point of appli- 
cation and decompose the forces into two others, one of which lies 
in the plane, and the other at right angles to it. The components 
thus obtained, which lie in the plane, are combined according to 
the rule given in the last paragraph, so as to give a single result- 
ant, and those at right angles to the plane give, by simple addition, 
another resultant. From these two components, which are at right 
angles to each other, we find the resultant according to the well- 
known rule (§ 77). 

This method of proceeding is graphically represented in Fig. 

95. MF, = Pi, MP, =: F„ M P^ = P3 are the simple forces, 
A B i^ the plane (plane of projection) and ^^is the axis at right 
angles to it. From the decomposition of the forces Pj P2, etc., we 
obtain the forces 81 S2, etc., in the plane, and the forces iVj iV",, etc., 
along the normal Z Z. The former are again decomposed into the 
components §1, ft? etc.. Pi, P2, etc., which, by addition, give the 
resultants Q and P, from which, as components, we determine the 
resultant 8., which, combined with the sum of all the normal forces 
Nx N^, etc., gives the required resultant P. 

If we put the angles of inclination of the directions of the 
forces to the plane equal to jSj, (5.2, etc., we obtain for the forces in 
the plane Sx = Pi cos. (3^, S^ = Pa cos. (3^, etc., and for the normal 
forces JVi = P, si^i. /3i, ]Vq = P^ sin. 13^, etc. Designating the angles 
which the projections of the directions of the forces in the plane 



§81.] 



MECHANICS OF A MATERIAL POINT 



IS.'^ 



A B form with the axis ^ Xby a^, a^, etc., that is, putting /S'l If X 
= aj, 8i MX — ao, etc., we obtain the following three forces, which 



Fig. 95. 




lorm the edges of a rectangular parallelopipedon (parallelopipe- 
don of forces) : 

Q — 8iCOS. Oj 4- Si COS. ao + . . ., or 

1) Q = P^ COS. i3i COS. «! + Pg COS. 13.2 cos. ag + . . ., 

2) R = Pi cos. i^i sin. a^ + Pg cos. 13^ sin. a.^ . . . and 

3) ]\^ =1 P^ sin. [3i + Po sin. 13^ + ... 

From these three forces we obtain the final resultant 

4) P = V~Q' + P^ + JV% 

and its angle P M S = (3 of inclination to the plane of pro> 
jection by the aid of the formula 

^ VQ' + R' 
Finally, the angle 8 M X= a, which the projection of the re- 
sultant in the plane A B forms with first axis X X, is given by 
the formula 

6) tang, a ~ ~, 

If A„ Ag, etc., are the angles formed by the forces Pi, P^ with the 
axis M X, Wj, |[/o . . ., the angles formed by them with the axis 
M Y and v,, v^, etc., the angles formed by them with the axis M Z, 
ire have also 



184 



GENERAL PRINCIPLES OF MECHANICS. 



[§81. 



5*) COS. X = ^, COS. pi = -^, COS. V — y^f 



1*) Q = P^ COS. Ai 4- Ps COS. ^2 + . . ., 
2*) E = Fi COS. fix + Pi COS. jLts + . . . and 
3*) J^= Pi COS. V, + P2 COS. v<, + , . . 
The value of the resultant is given by the formula 

4*) P = V Q' + R''^^^W% 
and the direction of the same by the formulas 

Q R N 

p^, COS. ^ = -p, COS. '^ ^ pT 

in which A, fi and v denote the angles formed by P with the axes 
MX,MY,MZ. 

"We have also cos. A = cos. a cos. (3, cos. 11 = sin. a cos. /3, and 
1; = 90" — (3, or COS. v z= sin. 13. 

Example. — In order to raise vertically a 
weight (x, Fig. 96, I and II, by means of a 
rope passing over a fixed pully, three work- 
men pull at the end of the rope A with the 
forces Pj = 50 pounds, Pg = 100 pounds and 
Pg =: 40 pounds ; the directions of these 
forces are inclined at an ansrle of 60° to the 
horizon, and form the horizontal angles 
' 8^ A 82 = S^ A 8^ = 185° and 8^ A 8^ = 
90° with each other. What is the inten- 
sity and direction of the resultant which we 
can put equal to the weight G, and how 
great could this weight be made, if the forces 
had the same direction ? 

The vertical components of the forces are 
I^i=Pi sin. /?i=50 sin. 60°=43,30 pounds, 




m 



iPg sin. 



3., 



and iV^3 - 



Po sin. j3. 



100 sin. 60° = 86,60 pounds 
3 ..,.. jB^ = 40 sin. 60° = 34,64 
pounds; consequently, the vertical force is 
N = N^ -{- N^ + N^ = 164,54 pounds. 

The horizontal components are 
8^ = Pj COS. iS^ = 50 COS. 60° = 25 pounds, 
/S'g = Pg COS. /^g = 100 COS. 00° = 50 pounds 
and 8^ =Pz ^^- 1^3 — 40 cos. 60° = 20 pounds. 

If we pass an axis X X in the direction 
of the force 8-^ , we have for the component 
forces in this direction 

Q= Qi + Qz+Qs^S^ COS. a^ -f 82 cos. Cg + 
8s COS. og = 25 COS. 0" + 50 cos. 135° + 
20 cos. 270° = 25 . 1-50 . 0,7071-20 . = 
25 — 35,355 = — 10,355 pounds, and for the 
comiDonent in the direction Y T 



§82.] 



MECHANICS OF A MATERIAL POINT. 



185 



R = R^ +i?3 + jKg == /S'j dn. a^ + S^ sin. a^ + 8. 



3 sin. Cg = 25 dn. 0" + 



50 8in. 135° + 20 sin. 270° = 50 . 0,7071 
horizontal resultant 



20 = 15,355 pounds, and for the 



8 =^Q' + R'=:^/' 10,355^ + 15,355'^ = 18,520 pounds. 

The angle a, formed by this resultant with the axis X X, is determined 
by the formula 



tang. a= -— = — 



15,355 

The final resultant is 



= — 1,4828, whence a = 180°— 56° = 124°. 



P= ■VN''+ 8' = ^/ 164,54^^ + 18,520^ = 165,58 pounds. 
The angle of inclination of this force to the horizon is determined by 



the formula 
tang. 13 = 



iV^ 164,54 



= 8,8848, whence we have {3 = 83° 35'. 



8 18,520 

If all the forces acted in the same direction, the resultant would be = 
50 + 100 + 40 = 190 pounds, or 190 — 165,58 = 24,42 pounds greater 
than the one just found. 

§ ^2, Principle of Virtual Velocities. — Erom the fore- 
going rules for the composition of forces, two others can be 
deduced, which are of great importance in their practical appli- 
cations. Let M, Fig. 97, be a ma- 



FiG. 97. 




terial point, M I^ — P^ and M E 
= Pq the forces acting upon it, 

P the resultant of the 
If we pass 



and if P 

forces Pi and Pg. 
through M two axes M X and 
ilSfl^ at right angles to each other, 
and decompose the forces P, and 
Ps, as well as their resultant P, 
into their components in the di- 
rection of these axes, i.e., Pi into §,, 
and Pi, Pi in Q.^ and P., and P into 
Q and P, Tve obtain the forces in the 
direction of one axis ^i, Q^ and ft and those in the direction of the 
other P, Pi and P2, and we have § = (>i + (>2 and P = Pi + Pa- 
If from any point in the axis Jf X we let fall the perpendiculars 
Pi, Li and L upon the directions of the forces P„ P2 and 
P, we obtain the right-angled triangles M L^ MO L.2, and M L, 
which are similar to the triangles formed by the three forces, viz., 

A if Pi O) A if P, ft, 
A 3f L, cp i^ M Pi ft, 
^ M L cp /\ M P Q, 



186 GENERAL PRINCIPLES OF MECHANICS. [^83. 

in consequence of this similarity we nave j^ I.E., ^ = jf-t^i 

p ~ wTT) ^^^ T> ~ ITT) ' substituting these values of Qx, §2 and 
Q in the formula Q = Qi + Qi, we obtain 



P . ML = P, . ML, + P, . ML,, 
In like manner we have 

R, _ OL, R, _ OL, R _ L 

p,~ M a p,~ M p ~ M a 

whence 



P. L = P,. L, ■\- P,. L,. 
The formulas hold good, when P is the resultant of three or 
more forces P,, P,, P^, etc., since we have, in general, 

Q= Qi + Q-2 - Q, + •" 
R = R, + R, -h R3 + ... 
We can, therefore, put, in general, 

1) P .~ML = P, . ML, + P, . Ml, + Pz . ML, + . , ,, 

2) P . 0L = P,. 0~Lr + P, . '0~L, + P3 . 0~L~, + . . . 

The resultant P of the forces Pi, P„ P3, etc., must correspond 
to both these equations, and they can therefore be employed to de- 
termine P. 

The first of these two formulas can also be employed for a sys- 
tem of forces in space, iV^ Q, R, Fig. 95, since here we have also 
iV = iVi + iVs + iVs + . . . , or 

P COS. V = p^ COS. v^ + P, COS. V, + P3 COS. V, -{- . . ., and also 

P . ~M'0 COS. p = P, . MO COS. v^ + P, M cos. v, + P, M cos. 1^3 + . . . 

§ 83. If the point of application M, Fig. 98 and Fig. 99, moves 
to 0, or if we imagine the point of application moved forward 

Fig. 98. Fia. 99. 





through the space M = cc, we call the projection M L — s of 
this space x upon the direction of the force M P the space described 
by the force P, and the product P 5 of the force by the space is the 



§84.] MECHANICS OF A MATERIAL POINT. 187 

work done dy the force. If we substitute these quantities in the 
equation (1) of the last paragraph we obtain 

hence the loorlc done 'by the resultant is equal to the sum of the ivorh 
done hy the component forces. 

In adding the mechanical effects we must, as in adding the 
forces, pa}^ attention to the signs of the same. If one of the forces 
Qif §2> of the foregoing paragraph, acts in the opposite direction to 
the others, then it must be introduced as negative quantity ; this 
force, as for example, ft in Fig. 94, § 80, is, however, a component 
of a force P^ which, under the circumstances supposed in the fore- 
going paragraph, opposes the motion M L^ of its point of applica- 
tion; we are, therefore, obliged to treat the force P, Fig. 99, which 
acts in opposition to the motion ML, as negative, if we consider 
the force P, Fig. 98, which acts in the direction of the motion M L, 
to be positive. 

If the forces are variable, either in magnitude or in direction, 
then the formula 

P 5 = Pi 5i + P2 5-2 + P3 53 + . . . 

is correct only for an infinitely small space s, 5i, ^2, etc. 

We call the infinitely small spaces ctj, a.^, a.^, etc., described by the 
forces corresponding to the infinitely small space described by the 
material point, the virtual velocities (Fr. vitesses virtuelles, Ger. 
virtuelle Greschwindigkeiten) of the same, and the law correspond- 
ing to the formula P a = P^ a^ -^ Pc^ a.^ 4- P3 0-3 is known as the 
principle of virtual velocities. 

^^ § 84. Transmission of Mechanical Effect. — According to 
the principle of ^4s viva for a rectilinear motion the work (P s) 
done by a force (P), when the velocity c of a mass M is changed 
into a velocity v, is 

Now if P is the resultant of the forces Pi, P2, etc., which act 
on the mass M, and if the spaces described by them are Si, s^, etc., 
while the mass M describes the space s, we have, from the forego- 
ing paragraph, 

P 5 = Pi 5i + P2 ^2 + . . . , 

from which we deduce the following general formula, 



p. s, + P, s, + . . . = (--3^) M 



188 



GENERAL PRINCIPLES OF MECHANICS. 



[§84. 



therefore the sum of the work done hy the single forces is equal to 
half the increase of the vis viva of the mass. 

If the velocity during the motion is constant, i.e., iiv — c and 
the motion itself is uniform, we have v' — c^ = 0, and therefore 
there is neither gain nor loss of vis viva, whence 
FiS, + F,s, + PsSs +...= 0; 
and the sum of the mechanical effects of the single forces is null. 

If, on the contrary, the sum of the mechanical effects is null, 
then the forces do not change the motion of the body in the given 
direction ; if the body hag no motion in the given direction, it will 
not have any imparted to it in this direction by the action of the 
forces ; if it had before a certain velocity in a given direction, it will 
retain the same. 

If the forces are variable, the variable velocity v can, after a cer- 
tain time, become the initial. This phenomena occurs in all peri- 
odic motions, which are very common in machinery. But v = c 

(v^ — c'\ 
— - — 1 Jf = 0, and therefore the gain or 

loss of mechanical effect during a period of the motion is = 0. 

Example. — A wagon, Fig. 100, weighing G = 5000 pounds is moved 
forward on a horizontal road by a force P^ = 660 pounds, inclined at an an- 
gle a — 24° to the horizon, 
Fig. 100, and is obliged to overcome 

^^ a horizontal resistance Pg 

= 450 produced by the fric- 
tion, what work must the 
force Pj do, in order to 
change the initial velocity 
of 2 feet of the wagon into 
a velocity of 5 feet ? 

If we put the space de- 
scribed by the wagon M 
= s, we have the work done 
by the force P^ 

= Pi . ML = P^8C0s. a = 660 . 8 cos. 24° = 602,94 . s, 
and the work done by the force Pg acting as a resistance is 

^(-Pg) .8=-450.s, 
consequently the work done by the motive force is 
P s = P^ 8 COS. a — P^ 8 COS. = (602,94 — 450) s = 152,94 s foot pounds. 

The mass, however, absorbed during the change of velocity the me- 
chanical effect 



P--^ 




85.] 



MECHANICS OF A MATERIAL POINT. 



189 



^-^ j a = l^—) ' 5000 = 0,0155 . (35-4) . 5000=1637,5 foot-pounds; 

putting the two effects equal to each other we obtain 153,94 . s = 1637,5, 
whence the space described by the wagon is 

1637,5 



MO = s = 



10,64 feet, 



153,94 
and finally the mechanical effect of the force P^ is 

Pj scos. a = 603,94 . 10,64 = 6415 foot-pounds. 

§ 85. Curvilinear Motion. — If we suppose the spaces {a, o^, 
etc.,) infinitely small, we can apply the foregoing formulas to cur- 
vilinear motion. Let M 8, Fig. 101, be the trajectory of the 

material point, and M P ^^ P 
the resultant of all the forces act- 
ing upon it. If we decompose 
this force into two others, the 

one of which M K = JT is tan- 



FiG. 101. 



M o 



■^K 




gent and the other M N =^ N 
normal to the curve, we call the 
former the tangential and the 
latter the normal force. 

While the material point de- 
scribes the element if = ct of its curved path M S, and its 
velocity changes from c to v^, the mass M absorbs the mechanical 

-^-^ — 1 M, during the same time the tangential force K 

performs the work E a, and the normal force the work JV . = 0, 
and consequently we have 

If, while the point describes the space M 8 = s = n a, the 
tangential velocity changes from c to v, and at the same time 
the tangential force assumes successively the values Ki, Kc^, . . K„y 
then 

and the work done is 



ir, + z; -f . . + K^ 



n 



)'=f^)« 



A = Ks 



r^) '. 



when K = 



K,-hE,-h,. + ir„ 



2 / ' n 

denotes the mean value of the variable tangential force. 



If we put the projection of the elementary space M = a upon 



190 GENERAL PRINCIPLES OF MECHANICS. [§85. 

the direction M L of the force = ^, we have also P ^ = K c, if, 
therefore, while the point describes the space M S — s = 7i a 
the resultant P assumes successiyely the values P^, P^ . . . P^, the 
projections of the elementary spaces are successively ^i, 1^2 .. . ^'„, 
and we have also 

and therefore 

When the direction of the force P remains constant, the pro- 
jections ^1, ^2 • • l« of the portions cr. a . . of the space or that of 
the whole space s = n a form a straight line 

31 B= x = ^, + ^, + .,^,. 

If we put ^ = m 1^, we can also write 

A = (P^ + P, + .. + P^)^={P, + p, + ... + PJ ~=Px, 

P + P + . . . + P 

where P denotes the mean — '— ^ of the forces, which 

m 

correspond to the equal portions ^ = — of the projections of the 

path on the direction of the force. 
We have, therefore, also 



Px= I ^ )m= (h-Jc)G, 

X -V / 



in which k denotes the height due to the initial velocity c and h 
that due to the final velocity v, and G the weight M g of the 
moving body. 

Therefore, in curvilinear motion, the entire ivorh done is equal 
to the product of the iveight of the hody moved and the difference 
of the heights due to the velocities. 

Remark. — The formulas, thus obtained by the combination of the prin- 
ciple of vis viva with that of virtual velocities, are particularly appli- 
cable to the cases of bodies, which are compelled to describe a given path, 
either because there is a support placed under them, or because they are 
suspended by a string, etc. If such a body is impelled by gravity alone? 
then the work performed by its weight G in descending a distance, whose 
vertical projection is s, is = G s, whence 

G s — Qi — Ic) G, I.E. s = h — h 



85.] 



MECHANICS OF A MATERIAL POINT. 



191 



Whatever may be the path on which a body descends from one hori- 
zontal plane A B^ Fig. 102, to another horizontal one G 2>, the difference 




of the heights due to the velocities is alw^ays equal to the vertical 
height of descent. Bodies, which begin to describe the paths E F, E^ F^, 
E^ i^25 ^t^M with equal velocities (c), arrive at the end of these paths with 
the same velocity, although they require different times to acquire it. 

If, for example, tL- mitial velocity is c == 10 feet, and the vertical height 
of fall = s = 20 feet, or A = s + ^ = 20 + 0,0155 . 10' = 21,55 feet, we have 
for the final velocity 

V = ^/¥gli = 8^,025 V21,55 = 37,24 feet, 

whatever may be the straight or curved line in which the descent takes 
place. / /' 



a 



THIRD SECTION. 



STATICS OF RIGID BODIES. 



CHAPTER I. 



GENERAL PRINCIPLES OF THE STATICS OF RIGID BODIES. 



§ 86. Transference of the Point of Application. — Al- 

though the form of every rigid body is changed by the forces which 
act upon it, that is, it is compressed, extended, bent, etc., yet in 
many cases we can consider the body as perfectly rigid, not only 
because this change of form or displacement of its parts is often 
very small, but also because it takes place during a very short 
space of time. For the sake of simplicity we will therefore con- 
sider, when nothing to the contrary is stated, a rigid body to be a 
system of points rigidly united to each other. 

A force P, Fig. 103, which acts upon a rigid body at a point A, 

transmits itself unchanged 

in its own direction JT X 
through the whole body, 
and an equal opposite force 
Pi will balance it, when its 
point of application A^ lies 

in the direction JT X, 
The distance of these 
points of application A 
and Ax from each other has no influence upon the state of equi- 
librium ; the two opposite forces balance each other, whatever the 
distance may be, if the points are rigidly connected. We can 



Fig. 103. 




§87,88.] 



STATICS OF RIGID BODIES. 



193 



therefore assert, that the action of a force P^ (Fig. 104) remains 
the same, no matter in what point A^, A^, A3, etc., of its direction 
it may be applied or act upon the body M. 

§ 87. If two forces Pi and Ps? Fig. 105, acting in the same 
plane are applied at different points A^ and ^2 to a body, their 

action upon the body is the same as if 
the point C at which the two directions 
intersect were the common point of ap- 
plication C of these forces ; for, accord- 
ing to the law just laid down, both 
points of application can be transferred 
to C without producing any change in 
the action of the forces. If, therefore, 
we make 



Fig. 105. 




C Q, = A,P, = P, and 

6 4f2 ^ A<i P<2 = P.2, 

and complete the parallelogram C Qi 
Q §2> its diagonal will give us the result- 
ant C Q = P 01 C Qi and C Q^ and also of the forces Pi and P2. 
The point of application of this resultant can be any other point A 
in the direction of the diagonal. 

If at a point B on the diagonal we apply a force B P = — P 

equal and opposite to the resultant A P — P, the forces Pi, Pa and 
— P will balance each other. 



§ 88. Statical Moment. — If from any point 0, Fig. 106, in 
the plane of the forces we let fall the perpendiculars Xj, L^ 
and L upon the directions of the component forces Pj and P2 
and of the resultant P, we have, according to § 82, 



p . L = p, . L, + p: . L,, 

and, therefore, from the perpendiculars or distances L^ and L^ 
of the components we can find that of the resultant by putting 



OL^ 



P,.0 L,-v P,.0 L, 
P 



While the intensity and direction of the resultant is found by 
means of the parallelogram of forces, the position L of the point 
of application is obtained by means of the last formula. 

13 



194 



GENERAL PRINCIPLES OF MECHANICS. 



[§88. 



If the directions of the forces, when sufficiently prolonged, form 
an angle Pj G P^ = a, the value of the resultant is 

1) P = VP,' +P/+ "^P.P.cos.a, 

If the direction of the resultant 
forms an angle POP]— a^ with the 
direction of the component P^, we 
haye 

P., sin. a 




2) sin. ay 



P 



If, finally, the distances from any 
point to the directions C Pj and C P^ 
of the giyen forces are Ly=^ a^ and 
Po = a.2, then the distance L ^^ a 
from this point to the direction O P 
of tne resultant is 

Pi a^ + Pa a.2 



3) a = 



P 



By the aid of the last distance a we can determine the position 
of the resultant without reference to any auxiliary point Chj de- 
scribing from with the radius a a circle, and by drawing a tan- 
gent L P to it, the direction of which is given by the angle a^. 

Example. — A body is acted w^on by the forces P^ =» 20 pounds and 
Pg = 34 pounds, whose directions form an angle P^ G P^ = a = 70° with 
each other, and their distances from a certain point are L^ = a^ =4 
feet and L^ = a^ = 1- foot ; what is the intensity, direction and posi- 
tion of the resultant ? The value of the resultant is 



P = V' 20^ + 34- + 2 . 20 . 34 cos. 70° = V 400 + 1156 + 1360 . 0,34203 
= l/ 2021,15 = 44,96 feet ; 

and its direction is determined by the angle a^, whose sine is 

34 . sin. 70° 



sin. Oj = 



-jj-^ — "5 hence log sin. fij = 0,85163 — 1, 



and the angle formed by the direction of the resultant with that of the 
force Pi isa^ = 45° 17'. The position or line of application of the result- 
ant is finally determined by its distance L from 0, which is 

20.4 + 34.1 114 ^^„, ^ . 
a = r—r,: = ^-^^ = 2,536 feet. 



44,96 



44,96 



g 89, 90.] 



STATICS OF RIGID BODIES. 



195 




§ 89. — We call the normal distances L^ ~ a^ and L^ = a^ 
of the directions of the forces from an arbitrary point 0, Fig. 107, 

the arms of the lever, or simply the arms 
(Fr. bras du levier, Ger. Hebelarme) of the 
forces, because they form an important ele- 
ment in the theory of the lever, which will 
be discussed hereafter. The product P a oi 
the force and the arm of the lever is called 
the statical moment of the force (Fr. moment 
des forces, Ger. statisches or Kraftmoment). Since P a — P^ax 
+ Pa ^2j the statical moment of the resultant is equal to the sum 
of the statical moments of the two components. 

In adding the moments, we must pay attention to the positive 
and negative signs. If the forces Pi and P2 act in the same direc- 
tion around 0, as in Fig. 107, if, e.g., the direction of the forces 
coincide with the direction of motion of the hands of a watch, they 
and their moments are said to have the same sign, and if one of 
them is taken as positive, the other must also be considered as 
positive. If, on the contrary, the two forces, as in Fig. 108, act in 

Fig. 108. Fig. 109. 





opposite directions around the point 0, they and their statical mo- 
ments are said to be opposite to each other, and when one is 
assumed to be positive, the other must be taken as negative. 

In the combination represented in Fig. 109 we have P a ~ 
P, ax — P2 ttoj since P^ is opposite to the force P^, or its moment 
Pa a<i is negative, while in the combination in Fig. 106 P a = 
Pi a-i + P., a-2. 

§ 90. Compcsition of Forces in the Same Plane. — If 

three forces Pi, Po, P3, Fig. 110, are applied to a body at three 
different points A^, Ae,, A3 in the same plane, we first combine two 

(Pi, P2) of these forces so as to obtain a resultant C Q = Q, and 
then combine the latter with the third force (P3) according to the 




196 GENERAL PRINCIPLES OF MECHANICS. [§91. 

same rule, constructing with D Ei = C Q and D R^ — A^ P^ the 
parallelogram D Ri R Rj- The diagonal D R is, the required re- 
sultant P of Pi, Psj and P3. It is easy to see how we must pro- 
ceed, when a fourth force P4 is added. 

Here the intensity and direction of the resultant is found in ex- 
actly the same manner as when the forces are applied at the same 

point (see § 80); the rules 
given in § 80 can be employed 
to calculate the first two ele- 
ments of the resultant, but 
the third element, viz., the 
position of the resultant or 
its line of application, must 
be determined by means of 
the formula for the statical 
moments. If Zi = ai, 
L2 = a.2, L-i = a-i and 
L ^= a are the arms of 
the three component forces 
Pi, Pi, P3 and of their re- 
sultant P in reference to an arbitrary point 0, and if Q is the re- 
sultant of Pi and P2 and K its arm, we have 

P a = Q . K + P3 a^ and Q . K — P^a^ -\- P., a^. 
Combining these two equations, we obtain 

P a = P^ a^ + Pi a.2 + P3 a-^, 
and in like manner when there are several forces 

P ^ = Pi CTl + P2 fi^2 + P3 «3 + • • • 

I.E., the {statical) moment of the resultant is always equal to the algS' 
hraical sum of the {statical) moments of the components. 

§ 91. If Pi, P2, P3, etc.. Fig. Ill, are the individual forces of a 
system, a^, a^, a^, etc., the angles Pi Pi X, P, D.^ X, P3 P3 X, etc., 
formed by the directions of these forces with any arbitrary axis 

JTXand a^, a.j, a-^, etc., their arms L^, L.j, Li, etc., in refer- 
ence to the point of intersection of the two axes X X and Y Y, 
we have, according to §§ 80 and 90, 

1) the component parallel to the axis X X 

Q = Pi cos. a, + Pa cos. a.2 4- . . ., 

2) the component parallel to the axis Y Y 

R = Pj sin, «! 4- P2 sin. a^ -\- ... 



§91.] STATICS OF RIGID BODIES. 197 

3) the resultant of tlie whole system 

P = V Q' + R\ 

4) the angle a formed by the resultant with the axis for which 

tang, a = -jy 

5) and the arm of the resultant or the radius of the circle to 
which the direction of the resultant is tangent 



a = 



Pi + Ps + . . . 




=x 



If h, bi, &2> etc., denote the distances D, D^, D^y etc., cut 

off from the axis X X, we have 

a — 1) sin, a, a^ = hi sin. a^, a^ = h^ sin. a^, etc., 
and therefore also 
_ P] Ji sin. a, + Pg ^2 sin. a^ + . . . _ R^hx + Pa ^2 + . . . 
P sin. a R 

If we replace the resultant (P) by ah equal opposite force (— P), 
the forces Pi, P^, P3 . . . (— P) will balance each other. 

If Xi, X. . . . and y^, yi - - • denote the co-ordinates of the points 
of application A^, Ao ... of the given forces P], Pa . . ., the mo- 
ments of the components of the latter are P, x^, Pq a^a . . . and Qi yi, 
Q^yi . . ., and the moment of the resultant is 

P a = [Ri Xi + R.i x^ -\- . , .) — {Q, yx + Qi y^ + . , .), 
and its arm is 



198 



GENERAL PRINCIPLES OF MECHANICS. 



[§91. 



a 



(BnX, + R^x.2 + . , .) — (Qi ^1 4- §2 2/2 + • • •) 



i/(^r+ e;+ ;. .y + (^r+ er + • . -y 

Example. — The forces Pj = 40 pounds, Pg = 30 pounds, P3 = 70 
pounds, Fig. 112, form with, the axis XX the angles a^ = 60°, Cg = — 80°, 
ag = 140°, and the distances between the points of intersection Z)^, Dg, Dg 
of the directions of the forces with the axis are B^JD^ = 4: feet, and B^ D^ 
— 5 feet. Required the elements of the resultant. The sum of the com- 
ponents parallel to the axis X X is 

Q = 40 COS. 60° + 30 COS. (- 80°) + 70 cos. 142° 
= 40 COS. 60° 4- 30 cos. 80° - 70 cos. 38° 
= 20 + 5,209 - 55,161 = - 29,952 pounds. 

The sum of those parallel to the axis Z Z is 

i? = 40 sin. 60° + 30 sin. (- 80°) + 70 sin. 143* 
= 40 sin. 60° — 30 sin. 80° + 70 sin. 38° 
= 34,641 - 29,544 + 43,096 = 48,193. 

Fig. 112. 




=X 



Hence it follows that the resultant 



P= VQ"" + P' = 1/29,952^ + 48,193^ = ^3219,68 = 56,742 pounds. 

The angle a formed by the latter with the axis is determined by the 
formula 



B 48,193 

tang, a = ^= - ^^^: 



1,6090, from which we obtain 



a = 180° — 58° 8' = 121° 52'. 
If we transfer the origin of the co-ordinates to Dg, we have the 
arm of the force 

-- Pj sin. a^ 5j + Pg sin. a^ h^ + . . . Pj &i + Pg Jg + . . . 



34,641 . (4 + 5) - 29,544 . 5 + _ 164,049 



56,742 



56,742 



P 

= 2,891 feet. 



§93.] 



STATICS OF RIGID BODIES. 



199 



and, on the contrary, the distance cut off on the axis X ^ 

OD = l = ^^^to = 3,404 feet. 
48,193 ' 

V 

§ 92. Parallel Forces.— If the forces P„ P„ P^, etc., Fig. 113, 
of a rigid system of forces are parallel, their arms Li, L^, 
Li, etc., coincide with each other; if through the origin we 

draw an arbitrary line XX, the directions of the forces will cut off 
from it the portions Di, Dj, D^, etc., which are proportional 
to the arms L^, L^, L^, etc., for we have a Dx Lx a^ ^ 
Dc^ Li CD ^ A Lz, etc. Designating the angle DxO Li=^ D^ L^, 
etc., by a, the arms Xi, L^, etc., by «i, a<i, etc., and the distances 
cut off Z>„ Do, etc., by Ji, J,., etc., we have 

^1 = hx COS. a, a^2 = ^2 cos. a, etc. 

Finally, substituting these values in the formula 



P a = Px Qx + P^ «2 + . . . , 



we obtain 



P h COS. a — Px bx COS. a -{■ P^h^ cos. a + . . . , 
or, omitting the common factor cos. a, we have 

Ph= Pxhx + P,h + ... 



Fig. 113. 



In every system of parallel 
forces we can substitute for the 
arms the distances D,, D2, 
etc., cut off' from any oblique line 
by the directions of the forces. 
Since the intensity and direction 
of the resultant of a system of 
forces with different points of 
application is the same as that 
of a system of forces applied in one point, the resultant of the sys- 
tem of parallel forces has the same direction as the components, 
and is equal to their algebraical sum ; hence we have 




1) 
3) 



P =: Pj 4- P, + P3 + . . . and 

Pi CTi + P2 «2 + • • • 



a = 



h = 



P, + P. + . . . 

Px hx -^ P, b, + . 

Pi + P, + . . . 



or 



200 



GENERAL PRIXCIPLES OF MECHANICS. 



[§93. 



Example. — The directions of the three forces P^ = 12 pounds, Pg = 
— '32 pounds and Pg = 25 pounds cut a straight line in the points D^, D^ 
and i>3, Fig. 113, whose distances from each other are D^ B^ = 21 inches, 
and I>2 i>3 = 30 inches ; required the resultant. The intensity of this 
force is 

P = 12 — 32 + 25 = 5 pounds, 

and the distance B-^ D of its point of application D in the axis XX from 
the point D^ is 

^ 12.0-32.21 + 35.(21 + 30) ^ 0-672 + 1275 ^ .^^^^^ 

5 o 

§ 93. Couples. — The resultant of two equal and opposite 
forces Pi and — Pi is 

P = P, + {- PO =. P, - P, z. 0, 
and its arm is 

a = = 00 (mnnitely great). 




Fig. 115. 




Ko finite force acting at a finite distance can balance a couple, 
but two such couples can balance each other. Let Pi and — Pj 
and —Pa and F^, Fig. 115, be two such couples, and Li = a^, M^ 
— L^ — L^ 3/i = a^ — hi, L. = cij and M^ = L^— L^. M^ 
= «2 — ^2 their arms measured from a certain point 0, then, when 
equilibrium exists, we have 

A Pi c?i — Pi («i — &i) — Pa ^2 + Po (a, — h.^ = 0, I.E. 
P, J, = P.3 J,. 

Two such couples balance each other when the product of one 
force by its distance from the opposite one is the same for both 
couples. 

A pair of equal opposite forces is called simply a couple (Fr. 
couple, Ger. Kraftepaar), and the product of one of its forces by 
their normal distance apart is called the moment of the couple. 



93.] 



STATICS OF RIGID BODIES. 



201 



Fig. 116. 



From the foregoing we see that two couples acting in opposite 
directions balance each other, when their moments are equal. 
That this rule is correct can be proved in the foll6wing manner. 
If we transfer the points of application of the forces Pi, Pa and 
— Pi,- P2 of the couples (Pi, - P,) and {P„ - P,), Fig. 116, 
to the points of intersection A and B of their lines of application, 

we can combine Pi and P2 
as well as — Pi and — P^ 
by means of the parallelo- 
gram of forces and obtain 
the resultants. If the di- 
rections of these resultants 
lie in the prolongation of 
the line A B, then these 
forces, and consequently the 
corresponding couples (Pi, 
- Pi), and (Ps, - P,\ bal- 
ance each other. If equilibrium exists, the triangle ABC formed 
\ij A B and by the directions of the forces — P and P2 must be 

similar to the triangles R A P^ and B R Pj, and consequently we 
have the proportion 

C B Pi 




CA 



-p- or the equation P^^C A — P^.G B. 



But the perpendiculars A L] = h^ and B Lc^ = i^ to the di- 
rections of the couples are proportional to the hypothenuses C A 
and C B of the similar triangles A C L^ and B C L^, and we can 
therefore put 

Pi h = P. h. 
The moments of two couples which balance each other are con- 
sequently equal to each other. 

If in the formula (§91) for the arm a of the resultant 



a — - 



Pi «i + Po «2 rt- . . . 



P 



we substitute P = 0, while the sum of the statical moments has a 
finite value, we obtain a = 00, a proof that in this case there can 
be no other resultant than a couple. 

If the forces of a system shall balance each other, it is necessary 

not only that the resultant P = l^^^ + P"" of the components Q 
and P, but also that its moment 

P a = P^ Oi + P2 «2 + • • • shall be = 0. 



202 



GENERAL PRINCIPLES OP MECHANICS. 






Example. — ^If one couple consists of the forces P^ = 25 pounds and 

— 11= — 25 pounds and the other of the forces Pg = 18 pounds and 

— Pg = — 18 pounds, and if the normal distance between the first couple 
Is 5 = 3 feet, then to produce equilibrium»it is necessary that the normal 
distance or arm of the second couple shall be 

h = -j^ = H feet. 
f 
§ 94. Composition and Decomposition of Couples. — The 

composition, and decomposition of couples acting in the same plane 
is accomplished by a mere algebraical addition, and is therefore 
much simpler than the composition and decomposition of single 
forces. Since two opposite couples balance ear^h other, when their 
moments are equal, the action of two couples is the same and the 
couples are said to be equivalent, when the moment of one couple 

is equal to that of the other. If, therefore, 
the two couples (Pi, — Pi) and (P.? — A), 
Fig. 117, are to be combined, we can replace 
the one (Pg, — P^) by another which has 
the same arm A B = biHS the former couple 
(Pj, — Pi), and can then add the forces thus 
obtained to the others, and thus obtain a 
single couple. If h is the arm (7 P of the 
one couple and {Q, — Q) the reduced couple, 
we have Q d^ = F^ i^, and consequently 




-P. 



h 



hence one component of the 



resulting couple is 



P, + § z= P, + 






and the required moment of the resulting couple is 

(Pi + Q)h = Pi h + P. K 
In same manner the resultant of three couples may be found. 
If Pi Ji, P2 l^, and P3 h be the moments of these couples, we 
can put 

P^l^— Q Ji and P^h = R h, or 



Q = 



P. I. 



and R = 



P.h 



l, h ' 

from which we obtain the resultant 

(Pi + (? -f R) I, = P,h, + P, h + P3 h' 
In combining these couples to obtain a single resultant we 
must pay attention to the signs, since the moments of the couples 



§95.] 



STATICS OF RIGID BODIES. 

« 



203 



Fig. 118. 



tending to tarn the body in one direction are positive, and the mo- 
ments of those tending to turn it in the other are negative. We can 
now adopt the following principle for indicating the direction of 
rotation of a couple. Let us assume arbitrarily a centre of rotation 
between the lines of application of the forces of a couple ; then if 
the couple tends to turn in the direction of the hands of a watch, 
the couple is to be considered as positive, and if in the other 
direction, as negative. 

The foregoing rule for the composition of couples is also appli- 
cable, when the forces act in parallel 
planes. If the parallel couples (Pi, 
- Pi) and (P<,, - P,),'Fig. 118, in 
the parallel planes M M and N N 
have equal moments Pi hi and 
Pa hi and act in opposite directions 
to each other, they will also balance 
each other ; for they give rise to two 
resultants Pi + Pg and — (Pi + 
Ps), which balance each other, as 
they are applied in the same point 
E, which is determined by the equa- 
tions 




E A,P, = E G,P,,E B.P, = E D.P,Q.n^ 



\- 



P, h, = P, h,, I.E. A B.P,= G D.P„ whence 
E A\E B'.A B = E G\ED'.CD\ 

hence this point coincides with the point of intersection of the two 
transverse lines A C and B D. 

Since the couple (Ps, — Ps) balances every other couple acting 
in a parallel plane with an equal and opposite moment, it follows 
that every couple can be replaced by another which has the same 
moment, and which acts in a plane parallel to that of the first. 

If, therefore, several couples whose planes of action are parallel 
are applied to a body, they can be replaced by a single couple whose 
moment is the algebraical sum of their moments, and whose plane, 
which in other respects is arbitrary, is parallel to the planes of the 
given system. 

§ 95. If two couples (P„ — P,) and (Ps, — PO act in two differ- 
ent planes E M E^ and F N F^, Fig. 119, whose line of intersection is 



304 



GENERAL PRIIj^CIPLES OF MECHANICS. 



[§95. 



the straight line A B, and which form with each other a given 

angle 

EA F= E,B F, = a 

we can, after having reduced them 
to the same arm A B, combine 
them by means of the parallelo- 
gram of forces. We obtain thus 
from Pi and P2 the resultant R, 
and from — Pj and— Pg the result- 
ant — R. These two resultants 
being equal and opposite, form 
another couple, whose plane is 
given by the direction of R and 
- P. 

The resultant R can be found 
according to § 77 by means of the 
formulas 




R = 

sin. /3 = 



V F,' -{- P,' + 2 P, P, COS. a and 
Pg sin. a 



R 



in which (i denotes the angle E A R = Ei B R formed by the 
direction of the resultant with that of the component Pj. If the 
arm is A B = c^ and if we put the moment P^ c = P a and the 



moment P<^c =■ § J or Pi = 



Pa 



and P^ = 



Qh 



we obtain 



R 



=/( 



Pa 



+ 



Qh 



V2^^ 



Qb 



COS. a, 



C I \ C j c c 

or the moment of the resultant of the couples (P, — P) and 

(ft ~ Q) 

Rc= V{Pa)' + {Qby +2 P a.Qi.cos.a, 

and in like manner for the angle formed by its plane with that of 
the first couple (P, — P) we have 



sin. 13 = 



Re 



sin. a. 



We can therefore combine and decompose couples acting in the 
different planes in exactly the same manner as forces applied at the 
same point, by substituting instead of the latter the moments of 
the former, and instead of the angles, which the directions of the 
former make with each other, those formed by their planes of action. 



§96.] 



STATICS OF RIGID BODIES. 



205 




The referring back of the theory of couples to the principle of 
the decomposition of simple forces can be greatly simplified by in- 
troducing the axis of rotation instead of the plaue of rotation of 
the couple. We understand by the axis of rotation or axis of a 
couple, any perpendicular to its plane. Since every couple can be 
arbitrarily displaced in its plane without changing its action upon 
the body, we can pass the axis of the couple through any given 
point. 

Since the plane and the axis of a couple are at right angles to 

each other, the axes A X, A Y 
and A Z, Fig. 120, form the same 
angles with each other as the 
planes A E K, A F K Sind AGK 
themselves. If one of the couples 
is the resultant of the other two, 
we see from what precedes, that 
the diagonal of the parallelogram 
constructed with the moments P a 
and Q h will give the moment 
of the resultant; if therefore we 
lay off upon the axes A X and A Y the moments P a and Q h, 
and then complete the parallelogram, we obtain in its diagonal not 
only the axis ^ Z of the resulting couple, but also its moment R c. 
We see, therefore, that couples are combined and decomposed in ex- 
actly the same ivay as simple forces, provided we substitute for the 
directions of the forces the axes of the couples and the moments 
of the latter for the forces themselves. All the rules for the com- 
position and decomposition of forces given in § 76 and § 77, etc., 
are in this sense applicable to the composition and decomposition 
^ of couples. 

§ 96. Centre of Parallel Forces. — If the parallel forces He 
in different planes, their composition must be effected in the fol- 
lowing manner. Prolonging the straight line A^ A^, Fig. 121, 
which joins the points of application of two parallel forces Pi and 
Pi, until it meets the plane which contains the axes MX and M Y, 
which are at right angles to each other, and taking the point of 
intersection K as the origin, we have for the point of application 
A of the resultant Pi + P2 of these forces 



(P, + P,) . KA = P, . KA, + P, . KA,, 



206 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 96. 



Now since B, B^ and B^ are the projections of the points of ap- 
plication A, A^ and A^ upon the plane X Y, we have 
AB:A,Br.A,B, = KA: K A, : K A„ 
and therefore also 



Fig. 121. 



(Pi + P,).AB = P,,A,B,-\- P,.A, B,. 
If we designate the normal distances A^ B^, A^ B^, A^ B^, etc., 

of the points of application 

from the plane X X by z^, z^, 
Zz, etc., and the normal dis- 
tance of the point of applica- 
tion A from this plane by 2, 
we have for two forces 

(Pi + P,)z = P,z, + P^z,; 
and for three forces, since (Pi 
+ P2) can be considered as 
one force with the moment 
Jr^i Z\ + -t 2 ^2? 
(Pi + P2 + Pa) Z 

— P^Zi-\- Pa Z^-\-Pz Zz, etc. 

Consequently we have in general 

(Pi + P2 + P3 + . . .) ;2 z= Pi ^1 4- P2 ^2 + Ps^a. . ., 

and therefore 

-L\ Z\ + JT 2 ^2 1 • • • 




1) 



z = 



Pi + P2 + . . . 

If, in like manner, we denote the distances A G and A D of the 
point of application A of the resultant from the planes X Z and 
Y Zhj y and x, and the distances of the points of application A^, 
^2 • • • from the same planes by y^, y^ . . . and Xi^ Xc^ , . ,, we obtain 

_ Piyx + Pg ^2 H- » . . 
Pi + P2 + . . . 

, _ JL\ X\ -\- -t 2 "^2 ~r . . • 

6) 



2) y 



and 



X 



Pi + A + . . . 

The distances, x, y and z, from three fixed planes, as, e.g., from 
the floor and two sides of a room, determine completely the point A ; 
for it is the eighth corner of the parallelopipedon constructed with 
X, y and z ; hence there is but one point of application of the re- 
sultant of such a system of forces. 

Since the three formulas for x, y and z do not contain the angles 
formed by the forces with the fixed planes, the point of application 
is not dependent upon them or upon the direction of the forces ; 



97.] 



STATICS OF RIGID BODIES. 



207 



the whole system can therefore be turned about this point without 
its ceasing to be the point of application, as long as the forces re- 
main parallel. 

In a system of parallel forces we call the product of a force by 
the distance of its point of application from a plane or line the 
moment of this force in reference to the plane or line, and it is also 
customary to call the point of application of the resultant the ce7i- 
tre of loaraUel forces (Fr. centre des forces parall^les, Ger. Mittel- 
punkt des ganzen Systems). We obtain the distance of tlie centre 
of a system of parallel forces from any plane or line (the latter, when 
the forces are in the same plane) by dividing the sum of the stati- 
cal moments by the sum of the forces themselves. 



Example. — If the forces are 


Pn 


5 


- 7 


10 


4 pounds. 


and their distances or the co- 
ordinates of their points 
of appHcation are 




1 

2 
8 


2 
4 
3 



5 

7 


9 feet. 
3 " 
10 " 


we will have the moments 


Jr^ X^ 
PnVn 


5 

10 


- 14 

- 28 



50 


36 foot pounds. 
12 " 


-^n ^n 


40 


— 21 


70 


40 " 



Now the sum of the forces is = 19 — 7 = 12 pounds, and therefore 

the distances of the centre of parallel forces from the three co-ordinate 

planes are 

5 + 36 



X 



y 



z = 



14 27 



9 



12 


"12 


10 + 50 + 12 - 


-28 


12 




40 + 70 + 40 - 


-21 



= T^ = 4- = 3,25 feet, 



12 



44 
12 
129 
I2" 



= :^ = — = 3,66 feet, and 

43 
= — = 10,75 feet. 
4 ' 



V 



97. Forces in Space. — If we wish to combine a system of 
forces directed in different directions, we pass a plane through 
them and transfer all their points of application to this plane, and 
then decompose each force into two components, one perpendicular 
to and the other in the plane. If (i^, (3,... are the angles formed 
by the directions of the forces with the plane, the components nor- 
mal to the plane are P, sin. (3^, P^ sin. (3^... and those in the plane 
are Pi cos. /?„ Pa cos. (ic^, etc. The resultant of the latter can be ob- 
tained as indicated in § 91, and that of the former as indicated in 



208 



GENERAL PRINCIPLES OF MECHANICS. 



[§ ^J7. 



the last paragraph. Generally the directions of the two resultants 
do not cut each other at all, and the composition of the forces so 
as to form a single resultant is not possible. If, however, the re- 
sultant of the parallel forces passes through a point K, Fig. 122, 
in the direction A B of the resultant P of the forces lying in the 
plane (that of the paper), a composition is possible. Putting the 
ordinates of the points of application K of the first resultant C 
— D K — u and D — G K = v,i\iQ arm of the other L — a 
and the angle B A formed by the latter with the axis X^,- a, 
then the condition for the possibility of the composition is 

u sin. a ■\- V cos. a =z a. 
If this equation is not satisfied, if, e.g., the resultant of the nor- 
mal forces passes through Ki, it is not possible to refer the whole 
system of forces to a single resultant, but they can be replaced by 



Fig. 122. 



Fig. 123. 









1 


' 








B 






h< 








5f 


D 


X 


y 




1 \ 


0^ 


7^ 


( 


: c. 


[0 




a resultant R, Fig. 123, and a couple (P, — P) by decomposing 
the resultant N of the parallel forces into the forces — P and P, 
one of which is equal, parallel and opposite to the resultant P of 
the forces in the plane. 

We can accomplish directly this referring of a system of forces 
to a single force and to a couple by imagining a system of couples, 
whose positive components are exactly equal in amount and direc- 
tion to the given forces, to be applied to the body at any arbi- 
trary point. These couples naturally do not change the state of 
equilibrium, for being applied at the same point they counteract 
themselves. On the contrary, the positive components can be 
combined according to known rules (§ 81) so as to give one result- 
ant, while the negative components form with the given forces 
couples, whose resultant (according to § 95) is a single couple. 
After these operations have been performed, we have only one force 
and one couple. 



98.], 



STATICS OF RIGID BODIES, 



209 



^ § 98. Principle of Virtual Velocities. — If a system offerees 
Pi, Pci, Ps, Fig. 124, which act in a plane, have a motion of trans- 
lation, that is, if all the points of application A^, A^, As describe 
equal parallel spaces Ai B^, A^ B^, A^ Bs, then (according to 
the meaning of § 81) the work done by the resultant is equal to 

Fig. 124. 




the sum of the work done by the components, and consequently, 
when the forces balance each other, this sum is = 0. If the pro- 
jections of the common space Ai B^ = Ac^ B^, etc., upon the di- 
rections of the forces are Ai Li, A^ L^, etc., = 5i, 5^, etc., the work 
done by the resultant is 

P S = Pi 5i + Pa Ss + . 

This law is a consequence of one of the formulas in § 91. Ac- 
cording to it, the component Q of the resultant parallel to the axis 
X X is equal to the sum 

of the components of the forces Pj, Po, etc., which are parallel to 
it. Now from the similarity of the triangles AxB^Lx and AiFiQi 
we kuow that 

ft __ ■^4^_Xi _ _s^ 

Pi A, B, A B' 

and therefore we have 

1 = "T-DJ ft = -T-D' ®*c. and Q = 



Hence, instead of 
we can put 

14 



A B' ^' A E 

C = ft + ft + .•• 

P S = Pi 5i + Ps ^2 + 



AB' 



21U GENERAL PRINCIPLES OF MECHANICS. [§ 99, 100. 

§ 99. Equilibrium in a Rotary Motion.— If a system of 
forces Pi, P„ etc., Fig. 125, acting in the same plane, is caused to 

turn a very small distance 
about a point 0, the principle 
of virtual velocities announced 
in § 83 and § 98 is applicable 
here also, as can be demon- 
strated in the following man- 
ner. According to § 89 the mo- 
ment of the resultant P . 0~L 
— P a \& equal to the sum 
of the moments of the com- 
ponents, or 

P a = Pi ai -^^ P^ a^i + . . . 
The space Ax Bi, corresponding to a rotation through a small 

angle AiO Bi= (3° or a small arc 13 = —— . n, is situated at right 

angles to the radius A-^, therefore the triangle Ai Bi C^ formed 
by letting fall the perpendicular P, Ci upon the direction of the 
force, is similar to the triangle A^Lx formed by the arm Li — <2i, 
and we have 

OLx _ Ax Cx 

OAx ~ Axb; 

If we put the virtual velocity Ax Cx =^ Ox and the arc Ax Bx 
= Ax . 3, we obtain 

Ax . O^ (7i n • TT ^2 




a, == 



-^ = ~, and in like manner a^ = t^, etc. 



Substituting these values of a^, a^, etc., in the above equation, 
we obtain 

^ - ^i^ 4- :^^ -!- etc 

(3 - f3 + [3 +--"^^^-^ 

or since (3 is a common divisor, 

P (J = P^ (7X + Pi (^2 + ' ' ', 

as we found in § 83. 

Therefore, for a small rotation, the work {P o) done by the re- 
sultant is equal to the sum of the work done by the components. 

§ 100. — The principle of virtual velocities holds good for any 
arbitrarily great rotation, when, instead of the virtual velocities 
of the points of application, we substitute the projections 



§ 101.] STATICS OF RIGID BODIES. 211 

Zi (7i, Zs Cs, Fig. 126, of the spaces described by the ends Zi, i/g, 

Fig. 126. 




€tc., of the perpendicnlars ; for multiplying the well-known equa- 
tion for the statical moment 

P a — P^a^ + P^ CL + . . . 
by sin. (3 and substituting in the new equation 

P a sin. 3 = Pj a^ sin. (i + P^ a^ sin. [3, 
instead of cti sin. [3, a.^ sin. 3 . . . the spaces 

B, sin. L, B, = D, B, = L, C, = s„ 

B2 sin. Lo Bo = B^ B^ = Z2 C^2 = s^, etc., 
we obtain 

P s = P, s, + P, s, + . . . 
This principle remains correct for finite rotations, when the di- 
rections of the forces revolve with the system, or when the point 
of application or end of the perpendicular changes continually so 
that the arms L^ = B^, etc., remain constant ; for from 

P a = Pi cii + Pi «o + . . ., 
by multiplying it by /3 we obtain 

P a (3 = P^ a^3 + P^a^ii + . . ., i.e., 

P s ^ P, s, + P, s, + . . ., 
when Si s.2, etc., denote the arcs Li Pi, L^ B.2, etc., described by 
the points of application Z,, P,,, etc. 

§ 101. A Small Displacement Referred to a Rotation. — 

Every small motion or displacement of a body in a plane can be 
considered as a small rotation about a movable centre as we will 
now proceed to show. Let A and P, Fig. 127, two points of the 
body (surface or line), be subjected to a small displacement, in con- 
sequence of which they now occupy the positions A^ and P,, A^ Pi 
being = A B. If we erect at these points perpendiculars to the 
paths A A^, and P P„ they will cut each other at a point C, about 
which we can imagine the spaces A J, and P Pi, considered as 
arcs of circles, to be described. But since A B = Ai B^, A C =. 



212 GENERAL PRINCIPLES OF MECHANICS. [§ 102. 

Ai O and B C — B^ C, the two triangles ABC and A^ B^ C 
are similar; the angle B^ C A^ i^ therefore equal to the angle 
B C A, and the angle of rotation A C A^ equal to the angle 
of rotation B G B^. K we make A^ Di = A Dwe obtain, since the 
angles A Ai C and D A C and the sides C A^ and C A are equal 
to each other, two equal, similar triangles C AiDx and GAB, 
in which G D, =^ G B smd Z A, G D, = Z A G B. Conse- 
quently, Z ^ C' ^1 is also = Z B G B^, and when the displace- 

placement of the line ^ ^ is small, 
every other point B of it will de- 
scribe an arc of a circle. Finally, if ^ 
is a point lying without the line A B 
but rigidly connected with it, the small 
space B Ei described by it can also 
be regarded as a small arc of a circle, 
whose centre is at (7; for if we make 
the angle B^ A^ B^ = E A B and the 
distance A^ E^ = A E, we obtain 
again two equal and similar trian- 
gles Ai G El and AGE, whose sides; 
G El and G E and whose angles 
Ai G El and AGE are equal to each other, and the same thing 
can be proved for every other point rigidly connected with A B, 
We can, therefore, consider any small motion of a surface or of a 
solid body rigidly connected with J. ^ as a small rotation about 
a centre, which is determined by the point of intersection G of the 
perpendiculars to the spaces A Ai and B Bi described by two 
points of the body. 

§ 102. G-enerality of the Principle of Virtual Velocities. 

— According to a foregoing paragraph (99) the mechanical effect 
of the resultant is equal to the mechanical effect of its components; 
for a small revolution of the system, and according to the last 
paragraph (101) any small motion can be considered as a revolu- 
tion ; the principle of virtual velocities is therefore applicable to- 
any small motion of a body or of a system of forces. 

If, therefore, a system of forces is in equilibrium, i.e., if the re- 
sultant is null, then after a small arbitrary motion the sum of the 
mechanical effects must be equal to 0. If, on the contrary, for a 
small motion of the body the sum of all the mechanical effects is 
equal to zerO; it does not necessarily follow that the system is in 




§ 103, 104.] CENTRE OF GRAVITY. 213 

equilibrium, for then this sum must be = for all possible small 
motions. Since the formula expressing the principle of virtual 
velocities fulfils but one of the conditions of equilibrium, in order 
that equilibrium shall exist it is necessary that this formula shall 
be true for as many independent motions as there are conditions, 
E.G., for a system of forces in a plane for three independent 
motions. 



CHAPTER II. 

THE THEORY OF THE CENTRE OF GRAVITY. 

f § 103. Centre of Gravity. — The weights of the different 
parts of a heavy body form a system of parallel forces, whose re- 
sultant is the weight of the whole body and whose centre can be 
determined by the three formulas of paragraph 96. We call this 
centre of the forces of gra"\dty of a body or system of bodies the 
centre of gravity (Fr. centre de gravite, Ger. Schwerpunkt), and 
also the centre of the mass of the body or system of bodies. If a 
body be caused to rotate about its centre of gravity, that point will 
never cease to be the centre of gravity, for if we suppose the fixed 
planes, to which the points of application of the single weights are 
referred, to rotate with the body, during this rotation the position 
of the directions of the forces in regard to these planes change, and 
on the contrary the distances of the points of application from 
these planes remain constant. Therefore the centre of gravity is 
that point at which the weight of a body acts as a force vertically 
downwards, and at which it must be supported in order to keep 
the body at rest. 

§ 104. Line and Plane of Gravity. — Every straight line, 
which contains the centre of gravity, is called a line of gravity, and 
every plane passing through the centre of gravity a 7:' toe of gravity. 
The centre of gravity is determined by the intersection of two lines 
of gravity, or by that of a line of gravity and a plane of gravity, or 
by the point where three planes of gravity cut each other. 

Since the point of application of a force can be transferred arbi- 
trarily in the direction of the force without affecting the action of 
the latter, a body is in equilibrium whenever any point of the ver- 
tical line passing through the centre of gravity is held fast. 



214 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 105. 



Fig. 128. 




If a body M, Fig. 128, be suspended at the end of a string C A, 
we obtain in the prolongation ^ ^ of this string a Hne of gravity, and 

if it be suspended in another 
way we find a second line of 
gravity/)^. The point of inter- 
section 8 of the two hnes is the 
centre of gravity of the whole 
body. If we suspend a body 
by means of an axis, or if we 
balance it upon a sharp edge 
(knife edge), the vertical plane 
passing through the axis or 
knife edge is a plane of gravity. 
Empirical determinations 
of the centre of gravity, such 
as we have just given, are seldom applicable ; we generally employ 
some of the geometrical methods, given in the following pages, to 
determine with accuracy the centre of gravity. In many bodies, 
such as rings, etc., the centre of gravity is without the body. If 
such a body is to be suspended by its centre of gravity, it is neces- 
sary to fasten to it a second body in such a manner that the cen- 
tres of gravity of the two bodies shall coincide. 

§ 105. Determination of the Centre of Gravity. — Let x^, 

Xci, Xs, etc., be the distances of the parts of a heavy body from one 
co-ordinate plane, y^, y^, y^, etc., those from the second, and z^, Zq, 
Zz, etc., those from the third, and let P,, Po, P^, etc., be the weights 
of these parts, we have, from § 96, for the distances of the centre 
of gravity of the body from the three planes 

-t J X\ -\- -L^ X^ -\- Jr^ X^ + ... 



X = 



y 



P,y, + P,y, + P^y, + .. 

Pi + P, + P3 + ... 
Pi 0] + P^z^ + P3 2:3 4- . . 



and 



P,-\-P, + Ps 

If we denote the volume of these parts of the body by Fi, Fg, 
F3, etc., and the weight of their units of volume by yi, y^, y^, etc., 
we can write 

Fi yi Xi + F2 ys ^^2 + F3 y3 ^3 + . ■ . 



X = 



Fyi + Fy, + F3y3 + . 



etc. 



§ 106.] 



CENTRE OF GRAVITY. 



215 



If the body is homogeneous, i.e., if y is the same for all the 
parts, we have 

( Fi a-i + F2 :C2 + . . .) y 
(F,+F + ...)y ' 

or, cancelling the common factor y, 



\) x = 



F + F + . . . 



2)y — — ^^ ^ -^^ ^^ , and 



d) z = 



F + F + . . . 
F ^1 + F2 2:2 + . . . 

"~Fr+~F + . . . • 



Consequently we can substitute for the weights of the different 
parts their volumes, and the determination of the centre of gravity 
becomes a question of pure geometry. 

AYhen one or two of the dimensions of a body are very small 
compared with the others, e.g., in the case of sheet-iron, wire, etc., 
we can regard them as planes or lines, and determine their centres 
of gravity by means of the last three formulas, substituting instead 
of the volumes F, F, etc., the surfaces F^, Fc,, etc., or the lengths 
'ij 'sj eic* 



Fig. 139. 



§ 106. In regular spaces the centre of gravity coincides with 
their centre, E.G., in the case of the cube, sphere, equilateral trian- 
gle, circle, etc. Symmetrical spaces have their centre of gravity in 
the axis or plane of symmetry. A body A D F H, Fig. 129, is di- 
vided by the plane of symmetry A B C D 
into two halves, which differ only in their 
position in regard to the plane, and the 
conditions are therefore the same on both 
sides of the plane ; the moments are con- 
sequently the same on both sides, and 
the centre of gravity is to be found in 
this plane. 

Since the axis of symmetry E F di- 
vides the plane surface A B F C D, Fig. 
130, into two parts, one of which is the 
reflected image of the other, the conditions are the same on each 




216 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 107. 



side ; consequently the moments on both sides are the same, and 
the centre of gravity of the whole surface lies in this line. 

Finally, the axis of symmetry K L oi Qi> body A B G H, Fig. 
131, is also a line of gravity of it ; for it is formed by the intersec- 





tion of two planes of symmetry A B C D and E F G H. 

For this reason the centre of gravity of a cylinder, of a cone and 
of a solid of rotation, formed by the revolution of a surface, or by 
being turned upon a lathe, is to be found in the axis of the body. 



Fig. 132. 

M 



§ 107. Centre of Gravity of Lines.— The centre of grav- 
ity of a straight line is at its centre. 

The centre of gravity of the arc of a circle A M B = b, Fig. 132, 
is to be found in the radius drawn to the middle M of the arc ; for 
this radius is an axis of symmetry of the arc. In order to deter- 
mine the distance C S=^/ oithe centre of gra\dty S from the cen- 
tre of the circle, we divide 
the arc into a very great 
number of parts and deter- 
mine their statical moment 

in reference to an axis X X, 
which passes through the 
centre C and is parallel to 
the chord A B = s. IfFQ 
is a part of the arc and F JS^ 
its distance from X X, its statical moment is = F Q . F JV, 
Drawing the radius F C = MC= r and the projection Q Rof F Q 
parallel to A B, we obtain two similar triangles F Q R and C F N, 
for which we have 




-X 



108.] 



CENTRE OF GRAVITY. 



217 



P Q:QR=C P'.P N, 

whence we obtain for the statical moment of an element of the arc 

P Q.PJSr=QR. GP=QR.r. 

But in the statical moments of all the other elements of the arc 
r is a common factor, and the sum of all the projections Q P of the 
elements of the arc is equal to the chord, which is the project ion of 
the entire arc ; consequently the moment the arc is = the chord ^ 
multiplied by the radius r. Putting this moment equal to the arc 
h multiplied by the distance y, oy b y = s r, we obtain 

y s sr 

r h ^ b 

The distance of the centre of gravity from the centre is to the rch 
dius as the chord is to the arc. 

If the angle subtended by the arc 5 is = i3° and the arc cor- 

responding to the radius 1 = (i = -—-^ tt, we have b = (3r and 



180^ 



i3 



s = 2r sin, ^, and consequently 

2 sin. ^ /3 . r 



y = 



a 
/? 



Fig. 133. 



For a semicircle i3 == rr and sin. ^ = 1, whence 

2 7 

^ = - r = 0,6366 ... r, approximatively = — r. 

§ 108. In order to find the centre of gravity of a polygon or 

combination of lines A B C D, Fig. 
133, we first obtain the distances of 
the centres I£, K, L of the lines 
AB=:l„ B C=h, OD = l,, etc.,. 
from the two axes X and Y, 
viz., H Hx = yx, H H<i = x^, K Kx = 
y^i, K K<i = x^, etc. The distances 
of the centre of gravity from these 
axes are 

Ix X\ ~r tj '^2 1" • • • 




8x 

OS, 



-.SS,=^x = 
SSx = y-^ 



'1 I" '2 I • • • 
?i yi + k y-2^ "^ 

t] "T" ti "T" . 3 • 



218 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 109. 



E.G., the distance of the centre of gravity S of a wire ABC, Fig. 
134, bent in the shape of a triangle from the base A B is 



jv^ = 2, = ii^i+i**- 



a + b h 



a + b -^ c a + b + c' 2' 



Fig. 134. 




A G 



when the sides opposite the angles 
A, B, C are denoted by a, h, c 
and the altitude C Ghj h, 

Ifwe join the middles^, K, M 
of the sides of the triangle and 
inscribe a circle in the triangle 
thus obtained, its centre will co- 
incide with the centre of gravity 
S ; for the distance of this point 
from one of the sides H Kis 



SB = NB 

A A B C 



,^ ^ 7i a -\- b h 

2 a + b ^ c 2 



ch 



2 {a -\- b -\- c) 
or constajit, and therefore — the dis- 



FiG. 135. 



a + b -h c 
tances S E and 8 F from the other sides. 

§ 109. Centre of G-ravity of Plane Figures. — The centre 
of gravity of a parallelogram A B C B, Fig. 135, is situated at the 

point of intersection 8 of its diagonals ; 
for all strips K B, formed by drawing 
lines parallel to one of the diagonals 
B B, Q:Te divided by the other diagonal 
A C into two equal parts ; each of the 
diagonals is therefore a line of gravity. 

In a triangle ABC, Fig. 136, every 
line C B drawn from an angle to the 
centre B of the opposite side ^ ^ is a line of gravity ; for it bisects 
every element K B of the triangle formed by drawing lines paral- 
lel to A B. If from a second angle A we draw a second line of 
gravity to the middle BJ of the opposite side B C, the point of in- 
tersection S of the two lines of gravity gives the centre of gravity 
of the whole triangle. 

Since BB = iBAaiidLBB=iBC, BBis parallel to ^ 
and equal to ^ A C, the triangle B E S i^ similar to the triangle 
C A 8 and G 8 = 2 8 B. Adding /S A we obtain C 8 + 8 B, 




§ no.] 



CENTRE OF GRAVITY. 



219 



I.E. C D = ^ 8D Qxidi inversely SD = I CD. The centre of 
gravity 8 is at a distance equal io \ C D from the middle D of the 
base and at a distance equal io % C B from the angle C. If we 
draw the perpendiculars C H and 8 Nto the base, we have also 



Fig. 136. 




Fig. 137. 




Ai CxSiDj 



]Sf 8 = I C H-y the centre of gravity 6^ is at a distance from the 
base of the triangle equal to one third of the altitude. 

The distance of the centre of gravity of a triangle ABC, Fig. 

137, from an axis X Xis 8 8,= B B, + J {C C, - B B,),hut 
B B, = I {A A, + B B,), and consequently we have 

y = 88, = lCC, + l.^, {A A, + B B,) = ^1A±_^_?1±.£^^ 



I.E., the arithmetical mean* of the di'stances of the angles from XX. 
Since the distance of the centre of gravity of three equal weights, 
appMed at the corners of a triangle, is determined' in the same way, 
the centre of gravity of a plane triangle coincides with the centre 
of gravity of these three weights. 

§ 110. The determination of the centre of gravity of a trape- 
zoid A B C B, Fig. 138, can be made in the following manner. 
The right line M N, which joins the centres of the two bases A B 
and C i>, is a line of gravity of the trapezoid ; for if we draw a great 
number of lines parallel to the bases, the figure will be divided into 
a number of small strips whose centres or centres of gravity lie 
upon the line M N. In order to determine completely the centre 
of gravity 8, we have only to find its distance 8 H from the 
base A B. 

Let the bases A B and C BhQ denoted by J, and h^ and the al- 
titude or normal distance between the latter by h. Now if we 
draw D E parallel to the side B C, vfe obtain a parallelogram 



220 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 110. 



B G D E, whose area is hi h and the distance of whose centre of 
gravity 8i from ^ ^ is == ^, and a triangle A D E, whose area is 

^ ^ and the distance of whose centre of gravity from A B 

h 
is -3. 



F- 




A O 



The statical moment of the trapezoid in reference lo A B 
is therefore 



Fy = l,h,~ + 



Ti (hi — hci) h h 



2 



= {b^ + 2 h) 



le 



• 3 ~ ^^ " "^ 6 ' 
but the area of the trapezoid is i^ = {bx + Jg) o> 

consequently the normal distance of the centre of gravity from 
the base is 

^ ^ ~ l{h + h)h ~ h + h, * 3' 
The distance of this point from the middle line K L = ^ ' 



2 



of the trapezoid is 






Ix + ^2 

In order to find the centre of gravity by construction, we have 
only to prolong the two bases, make the prolongation G G ^hi and 
the prolongation A F = Jg, and join the extremities i^and G thus 
obtained by a straight line ; the point of intersection S with the line 

M N is the required centre of gravity; for from US = l^ , ^ . ^ 
it follows that 

M S = ~ — -^i— . -7T— and N 8 — -, — r-^^ . —7^—, or 



M 8 
N 8 



hx + h, ' 3 •^' ^ hx + h,' ^ 

lx + 2h __ ^^bx-^h, _ M A+AF _ MF 
2 bx -V b,~ %x ^ \ b,~ G G ■\- N G~ N G' 




§111,112.] CENTRE OF GRAVITY 221 

which, in consequence of the similarity of the triangles M S F and 
N 8 Gyi^ perfectly true. 

If we denote by a the projection ^ of the side A D upon 
A B, the distance of the centre of gravity from the corner A is 
determined by the formula 

. rT_ _ W + 1x1), -\-K +a(ii +2^,) 

§ 111. In order to find the centre of gravity of any other four- 
sided figure A B C D, Fig. 139, we can 
divide it by means of the diagonal A 
into two triangles, and then determine 
their centres of gravity Si and 8^^ by 
means of the foregoing rules ; thus_we 
obtain a line of gravity 8i 8^. If we 
again divide the figure by the diagonal 
B D into two other triangles, and de- 
termine their centres of gravity, we 
obtain a second line of gravity, whose 
intersection with 8i 8^ gives the centre of gravity of the whole 
figure. 

We can proceed more simply by bisecting the diagonal A C ai 
M and laying off the longer portion B E oi. the other diagonal 
upon the shorter portion, so as to have D F — BE. We then 
draw F M and divide this line into three equal parts ; the centre 
of gravity is at the first point of division 8 from M as can be 
proved in the following manner. We have M 8x — \ M D and 
M 8i — i M B; consequently 8i 8^ is parallel to B D, but 8 8i 
multiplied hj a A C D ~ 8 8, multiplied by A A CB ov 8 8^. 
D E = 8 8,. BE, whence 8 8,: 8 8, = B E: D E. But we have 
B E = D F d.n^ D E = B F, consequently also 8 8^: 8 8, = 
D F\ B F. Hence the right line M F cuts the line of gravity 
Sx 8^ at the centre of gravity 8 of the -whole figure. 

^^ • § 112. If we are required to find the centre of gravity 8 of tk 
-polygon ABODE, Fig. 140, we divide it into triangles and find the 

• j statical moments in reference to two rectangular axes XX and Y Y. 
If the co-ordinates Ai = x^, Ac^ = y^, B^— x<i, B^ — 
. «/2, etc., of the corners are given, the statical moments of the tri- 
angles A B 0, B C 0, C D 0, etc., can be determined very simply 
in the following manner. The area of the triangle A B is, ac* 
cording to the remark which follows, = Z), = ^ (a^i ^ — i^a t/i)t 



222 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 11^^ 



that of the following triangle B C is =^ D^ = h [x^ y^ — x^ y^y 

etc., the distance of the centre of gravity of J. ^ from Y Y is, 
according to § 109, 

X\ ~\~ X^ ~r U Xi ~\~ X^ 



th =■ 



and that from XXiB= v^ = ^L+l.^ 
ity of the triangle B G are 



those of the centre of grav- 



u. 



x^ -T x^ , y^ -\- y% , 

= — 5 — and v^ — ^ ^ ^ , etc. 
o 6 



Multiplying these distances by the areas of the triangles we ob- 
tain the statical moments of the latter, and substituting the values 
thus found in the formulas 

DiUi + D<iUci + , . T Di Vi + i)2 t'o 4- . . . 
u — — ^r ^r and Vi = — :p, =c , 

we obtain the distances u = Si and v — S^ of the required 

centre of gravity S from the axes Y Y and X X. 

If we divide in two ways a polygon of n sides by means of a di- 
agonal into a triangle and a polygon of (^ — 1) sides, and then 
join the centre of the former with that of the latter, we obtain in 
this way two lines of gravity, whose intersection gives the centre 
of gravity. By repeated application of this operation, we can find 
by construction the centre of gravity of any polygon. 

Example. — A pentagon A B G B B, Fig. 140, is given by the co-ordi- 

Fig. 140. 




nates of its comers A, B, C, etc.. and the co-ordinates of its centre of 

gravity are required. 



§ 113.] 



CENTRE OF GRAVITY. 



223 



Co-ordinates 
given. 


Double area of the triangles. 


The triple co-ordi- 
nate of the centre 
of gravity. 


The sextuple statical 
moment. 

1 


X 


y 


3 ^« 


3 «^« 


6D,Ur, 


6 i>„ V,, 


24 

7 
-16 

— 12 

18 


II 
21 

15 

- 9 
—12 


24. 21 — 7. 11=427 

7 . 15 + 21 . 16=441 

16. 9 + 12. 15=324 

12 . 12 + 18 . 9^=306 

18. II +24. 12=486 


31 

- 9 

- 28 

+ 6 

+ 42 


32 

6 

— 21 

— I 


13237 

-3969 
-9072 

1836 

20412 


13664 

15876 

1944 

— 6426 

- 486 1 






Total, 1984 






22444 


24572 



The distance of thfi centre of gravity from the axis Y Yis therefore 



1 22444 ^^^^ 



and from X X it is 



S-j^ 



1 
^ = 3 



24572 

i98r 



= 4,128. 



Remakk. — If C A^ = x^, C B^= iCg, C A^ = y^ and G B^ ^=yz are the 
co-ordinates of two comers of a triangle A B C^ Fig. 141, the third comer 
C of which coincides with the origin of co-ordinates, its area is 

Fig. 141. J) = trapezoid ABB^A^ + triangle 

CBB^ — triangle GAA^ 




f-^) (^. 



^2) + 



«« 



Vz 



2 



^1 Vi _ ^1 yg — a^2 Vr 



"f 



3 2 

The area of this triangle is there- 
fore the difference between those 
of two other triangles C B^ A^ and 
(7 J. 2 B^, and one co-ordinate of 
one point is the base of one trian- 
gle and the other co-ordinate is the altitude of the second triangle. In 
like manner one co-ordinate of the second point is the altitude of the first 
>ytriangle and the other co-ordinate is the base of the second triangle. 

^ § 113. The Centre of Gravity of a Sector, A C B, Fig. 
142, coincides with centre of gravity S of the arc A^ Bi, which has 
the same central angle as the former and whose radius C Ax is two 
thirds of that CA of the sector ; for the latter can be divided by an 



224 



GENERAL PRINCIPLES OF MECHANICS. 



[§114. 



infinite number of radii into small triangles, whose centres of gravity 

are situated at a distance from the 
centre C equal to two thirds of ra- 
dius; the continuous succession of 
these centres forms the arc A-^ M\ B^, 
i The cdntre of gravity S of the sector 
lies, therefore, upon the radius which 
bisects this surface and at the distance 

^ ^ chord 2 ^9— r 4 sin. J, ji 

^ arc 3 3 (3 

from the centre, when r denotes the radius of sector and (3 the 
arc which measures its central angle A C B. 

For the semicircle [i — n, sin. -h 13 = sin. 90° = 1, whence 

4 . 14 

y — ^ — r = 0,4244 r, or approximatively ^ n 

For a quadrant we have 

44/1 44/2 

y = ^.j-^r = %^ r --= 0,6002 r, 




3*A TT 

and for a sextant 

4 h 



3 7T 
2 



n 



r 



0,6366 n 



Pig. 143. 



114. The Centre of Gravity of the Segment of a Circle, 

A B M, Fig. 143, is found by putting 
its moment equal to the difference 
of the moments of the sector A C B M 
and of the triangle A G B. If r is 
the radius G A, s the chord -4 B and 
A the area of the segment A B M, we 
have the moment of the sector 

= sector multiplied hj G Si — 

r . arc chord 




2 



arc 



2 1 , 
-r=^sr 



the moment of triangle 

= triangle multiplied by (7 /S^a = 77 r ^* T * q f 



2 



4 3 



r' 



4 



s r 



~ 3 12' 

and consequently the moment of the segment A 



§ 115.] 



CENTRE OF GRAVITY. 



225 



A,CS=Ay = ^sr' 



(ill - ^\ - 
\ 3 12/ ~ 



12' 



Hence the required distance is y = t^j^" 



For a semicircle s = 2 r and A =-tt r% ana therefore 



y 



Sr' 



12 



as we haye already found. 



TT r 
~2~ 



4 r 
3^' 




In the same way the centre of grav- 
ity 8 of B> section of a ring A B D B, 
Eig. 144, can be found; for it is the 
difference of two sectors A C B and 
D C E. If the radii are C A = r^ and 
G E = r<x and the chords A B =^ Sx 
and D E =^ Si, we have the statical 

3 S 

moment of the sectors -^^ and -^tt^, 



and consequently that of the portion of the ring 

^2 



2 2 

if = ^ — — , or smce - = — , 

3 Si Ti 



M = 



3 S 

^1 — ^2 



_£i 



The area of the piece of the ring is i^= 



/5 r,' (3 T, 



=H^). 



2 2 

in which ii denotes the arc which measures the central angle 
A C B ; hence the centre of gravity ^S'of the section of the ring is 
determined by the formula 

p q _ _M _ri^—r^^ 2 Si _ 2 /n' — rA cliord 



arc 



r^^ sin. ^ (3 



/3 



(1 + j^^ ( — 1 1 2 r, when r, — Tt 



F 

_ 4 sin. J i3 ^i' 

~ 3 ~~^ ' rT^^ 

= h and r^ + rs = 2 r. 

Example. — If the radius of the extrados of an arch is r^ = 5 feet, and 
that of the intrados is r^ = 3^ feet, and if the central angle is /3" = 130°, 
the distance of the centre of gravity of the front surface of the arch from 
ita centre is 



4 8in. 65" 


5« - 


-3,5^ 


4 . 0,9063 


125 - 42,875 


3,6252 


82,125 


3 arc. 130" ' 


5^ - 


- 3,5^ 


"" 3 . 2,2689 • 


25 - 12,25 


~ 6,8067 


. 12,75 


3,430 feet. 















15 



I 



226 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 115. 



y^ (§ 115.) Determination of the Centre of Gravity by the 
Aid of the Calculus. — The determination of the centre of gravity 
by means of the calculus is accomplished in the following man- 
ner. Let A N P, Fig. 145, he the given 
surface, A N=x\U abscissa and NP — y 
its ordinate. The area of an element 
of the surface is 

d F = y d X (see Introduction to the 
Calculus, Art. 29) and its moment in ref- 
erence to the axis of ordinates ^ ^T is 




i^from the axis A Y, 



M , d F — A N . d F = X y dx\ 
if we put the distance L 8 = A K of the 
centre of gravity >S' of the whole surface 
= U) we have 

Fu = f xy dx, 



and consequently 



. _^fxydx__fxydx 

-1.) U ^p; T, 9 . 

F J ydx 



Since the centre or centre of gravity M of the element N M P 
is at the distance N M — \y from the axis A X, the moment of 
d Fin reference to this axis ^ X is 



NM,d F= ly dF=\y''dx\ 

putting the distance K 8 — A L oi the centre of gravity 8 of the 
whole surface F from the axis A X, = v, we have 

F V — f ly"^ d X, and therefore 

' F ^ fydx' 

E.G., for the parabola, whose equation \^ y"" = j^ x ox y — Vp .x\, 
we have 

_ f Vp .x^ X dx _ Vp f x^ d X _ f x^ d X 



u = 



fVp.x^dx Vpfx^dx S^dx 



1^ 



3. /y» 



0YL8=AK~%ANy and, on the contrary, 



_ 1 f pxdx _ J 



v = i 



— 3 



Vj) f x^ d X 
'fpx = ly, 



— 5 



Vp' p 



/ »^ Cv fcv 



xh d X 



V^ 



X' 



a^ 



or 



K8= AL-%NP. 



116.] 



CENTRE OF GRAVITY. 



227 



Fig. 146. 




§ 116. The Centre of G-ravity of Curved Surfaces. — The 

centre of gravity of the curved surface (envelope) of a cylinder 

A B C D, Fig. 146, lies in the middle S of 
the axis if iV^ of this body ; for all the ring- 
shaped elements of the envelope of the cyl- 
inder, obtained by cutting the body parallel 
to its base, have their centres and centres of 
gravity upon this axis ; the centres of grav- 
ity form then a homogeneous heavy line. 
For the same reason the centre of gravity 
of the envelope of a prism lies in the middle 
of the line, which unites the centres of gravity of its bases. 

The centre of gravity 8 of the envelope of a right cone ABC, 
Fig. 147, lies in the axis of the cone one-third of its length from 
the base, or two-thirds from the apex ; for this curved surface can 
be divided into an infinite number of infinitely small triangles by 
means of straight lines (called sides of the cone). The centre of 
gravity of all these triangles form a circle H K, which is situated 
at a distance equal to two-thirds of the axis from the apex (7, and 
whose centre or centre of gravity S lies in the axis C M. 



Fig. 148. 





The centre of gravity of a- zone A B D E, Fig. 148, of a sphere, 
and also that of spherical shell, lies in the middle 8 of its height 
M N', for, according to the teachings of geometry, the zone has 
the same area as the envelope F G H K oi o, cylinder, whose height 
is equal to that M N of the zone and whose radius is equal to that 
C of the sphere, and this holds good even in the ring-shaped ele- 
ments obtained by passing an infinite number of planes parallel to 
the base through the zone ; hence the centre of gravity of the zone 
and of the envelope of the cylinder coincide. 

Remakk. — The centre of gravity of the envelope of an oblique cone or 



228 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 117. 



pyramid is to be found, it is true, at a distance from the base equal to one- 
third of the altitude, but not in the right line joining the apex to the 
centre of gravity of the periphery of the base, since by cutting the en- 
velope parallel to the latter we divide it into rings of different thicknesses 
on different sides. 

/ 
§ 117. Centre of Gravity of Bodies.— The centre of gravity 
S of a prism A K, Fig. 149, is the centre of the line uniting the 

centres of gravity if and iV"of the two bases 
A D and G K) for by passing planes parallel 
to the base through the body we divide it 
into similar slices, whose centres lie in M N, 
and whose continuous succession form the 
homogeneous heavy line M N, 

For the same reason the centre of gravity 
of a cyhnder is to be found in the middle of 
its axis. 

The centre of gravity of pyramid A D F, Fig. 150, lies in the 
straight line Jf ^joining the apex i^with the centre of gravity M 
of the base ; for all slices such as JV P Q R have, in consequence 
of their similarity to the base ABODE, their centre of gravity 
upon this line. 

Fig. 150. Fig. 151. 

F 






If the body is a triangular pyramid, like A B C D, Fig. 151, we 
can consider each of the four corners as the apex and the opposite 
side as the base. The centre of gravity is therefore determined by 
the intersection of the two straight lines drawn from the corners 
D and A to the centres of gravity M and N of the opposite surfaces 
^ ^ Cand B CD. 

If the right lines E A and E D are also given, we have (accord- 



§ 118.] 



CENTRE 01'' CllAVlTY. 



229 



Fig. 152. 




ing to § 109) E M ^ {E A and ENVIED. MJY is therefore 
parallel to A D and = ^ A B, and the triangle M W 8 is similar 

to the triangle DAS. In conse- 
quence of this similarity we have 
also M 8 =lDSoxD 8 =dM8 
SiiidMD = M8 ^ 8D = 4: M 8, 
or inyersely if ^ = | if i). The 
distance of the centre of gravity 
of a triangular pyramid from its 
base along the line joining the 
centre of gravity if of the base to 
the apex D of the pyramid is equal 
to one-fourth of this line. 

If the altitudes D H and 8 G 
are given and if we draw the line 
H M, we obtain the similar triangles D H M and 8 G M, in- wdiich, 
as we have just seen, 8 G = { D H. We can therefore assert that 
the distance of the centre of gravity of a triangular pyramid from 
its base is one-fourth and from its apex three-fourths of its altitude. 
Finally, since every pyramid and every cone is composed of tri- 
angular pyramids of the same height, the centre of gravity of every 
pyramid and of every cone lies at a distance from the base equal to 
one-fourth of the altitude and at a distance from the apex equal to 
three-fourths of the altitude. 

We determine the centre of gravity of a pyramid or of a cone 
by passing a plane, at a distance from the base equal to one-fourth 
the altitude, through the body parallel to its base and by finding 
the centre of gravity of this section or the point where a line 
drawn from the centre of gravity of the base to the apex will cut it. 

§ 118. If we know the distances A A^, B B^, etc., of the four 
corners of a triangular pyramid A B C D, Fig. 153, from a plane 
H K, the distance 8 8i of its centre of gravity 8 from the plane is 
their mean value 



8 8i =^ 



A A, + B B, + C C, ■¥ D D, 



which can be proved in the following manner. The distance of 
the centre of gravity M of the base ABC from this plane is (§ 109) 



MM, 



A A, + BB, + CO, 



230 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 118- 



and the distance of the centre of gravity S of the pyramid is 
S S, = MM, + i{DD,- MM,), 



Fig. 153. 




m which D D, is the distance of the apex. Combining the last twa 
equations, we obtain 

The distance of the centre of gravity of four equal weights 
placed at the corners of the triangular pyramid is also equal to tha 
arithmetical mean 

A A, + B B, + C C, -h D D, 

y = 4- 5 

consequently the centre of gravity of the pyramid coincides with 

that of these weights. 

Remakk. — The determination 
of the volume of a triangular pyra- 
mid from the co-ordinates of its 
corners is very simple. If we pass 
through the apex of such a, 
pyramid A B 0, Fig. 154, three- 
co-ordinate planes X Y,X Z, Y Z, , 
and denote the distances of the: 
corners J., B, C from these planes . 

we have the volume of the pyramid 
^3 2/1 ^8 (^1 Vs ^z + ^2 Vt ^a 




119.] 



CENTRE OF GRAVITY. 



231 



which is found by considering the pyramid as the aggregate of four ob- 
liquely truncated prisms. 

The distances of the centre of gravity of this pyramid from the three 
co-ordinate planes TZ^ XZ and X Fare 



X 



= Z ' y = Z ' and s _ J 



§ 119. The centre of gravity 8 of any polyhedron, such as 
A B G D 0, Fig. 155j can be found by calculating the statical 

/n^iA^^ Fig. 155. 

A Z 




moments and volumes of the triangular pyramids, such SisABCO, 
B C D 0, into which it can be decomposed. 

If the distances of the corners A, B, C, etc., from the co-ordinate 
planes Y Z,X Z and X Y, passing through the common apex of 
all the pyramids, are x^^ x^, x-s, etc., ?/„ y^, y^, etc., and z^, z^^, Zz, etc., 
we have the volumes of the various pyramids 
Fi = ± i (.Ti 2/2 Zz + x^ 1/3 z^ + Xz yi z^ — ^1 yz z^^ — x^y^ z^ — x^y.^ z^\ 
F2 = ± i (.^2 2/3 2^4 4- x^ 2/4 z^ + x^ y.2 Zz — x^ y^ z^ — x^ y^ z^ — x^ y^z^), 
etc., and the distances of their centres of gravity from the co-ordi- 
nate planes are 



Uy = 



Uci = -- 



^1 


+ 


X2 


+ 


X3 






4 






3/2 


+ 


X3 


+ 


x^ 



V, 



yi + y ^ + ys ^„ _ z^ + z, -{- z^ 



_ y-i + y^ + ^4 



02 + 2^3 + 2^4 



From these values we calculate the distances u, v, iv of the 
centre of gravity S of the whole body by means of the formulas 



232 



GENERAL PRINCIPLES OF MECHANICS. 



L§ lia 



u = 



VlUl -{- Vr,Ui + . .. 



V 



ViV, + F2 ^2 + . . . 



Fi ^(^i + Fg W2 + . . . 



w = 



F +F2 + ... 



5 and 



Example. — A body A B C D 0, Fig. 155, bounded by six triangles, is 
determined by the following values of its co-ordinates, and we wish to find 
the co-ordinates of the centre of gravity. 



Given Co- 
ordinates . 




The sextuple volume of the 

triangular pyramids; 
A B C and B C D O 


Quadruple 
Co-ordi- 
nates of the 
Centres of 
Gravity. 


Twenty-four fold 
Statical Moments. 


X 

20 


y 
23 


z 
41 






s 


5? 


§ 
^ 


24 


24 

Y n 'On 


1 
24 

Vn Wn 




[20.29.281 




[20.40.30^ 
















6K=\ 


23 30.12 


■ — - 


23.28.45 


' rr 31072 


77 


92 


99 


2S92544 


28n8624 


3076128 


45 


29 


30 




.41.45.40J 




141.12.29 J 
















12 


40 


28 




r 45.35.281 




[45.40.201 


■ 










i 

i 








6Ti=^ 


29.20.12 


- — - 


29.28.38 


' = 17204 


95 


104 


78 


1634380 


1789216 1341912 


38 


35 


20 




30.3840 




130.12.35. 














Total 48276 




4026924J4647840I4418040 



From the results of the above calculation we deduce the distances of 
the centre of gravity /S of the whole body from the planes YZ^X Z and X 7", 



u = ~r 



V = 



1 

4 



4026924 

48276 

464784 

48276 

4418040 

48276 



= 20,853, 
= 24,069, and, 

= 22,879. 



Remaek. — ^We can also determine the centre of gravity of a polyhedron 
by dividing it in two ways by means of a plane into two pieces and by 
joining the centres of gravity of each two pieces ; the intersection of the two 
lines gives the required centre of gravity. Since both lines are lines of 
gravity, the intersection must be the centre of gravity of the who! 3 body. 
If the body has a great number of corners, this process becomes very long, 
in consequence of the number of times this division must be repeated. 
The five-cornered body in Fig. 155, which must be divided in two ways 
into two triangular pyramids, has its centre of gravity at the intersection 
of the lines joining the centres of gravity of each two of these pyramids. 



120.] 



CENTRE OF GRAVITY. 



233 



Fig. 156. 



A 



/ / 



120. The centre of gravity of a truncated pyramid or frus- 
tum of a ijyramid A D Q N, Fig. 156, 
lies in the line Q M joining the centres of 
gravity of the two (parallel) bases. In or- 
der to determine the distance of this point 
from one of the bases we must calculate 
the volumes and moments of the complete 
pyramid A D F and of the portion N Q F, 
which has been cut away. If the areas 
of the bases A D and N Q are = Gi and 
Gi, and if the perpendicular distance be- 
tween them = h, the height x of the por- 
tion of the pyramid, which is wanting, is 
determined by the formula 

G, _ {Ji + xf 

G^ 







G 3i 



C D 



whence 



X 



'G, 



= \/^^orx = ~^= 



X' 

hVG, 



vg. 



and 



h -^ X ~ 



li V G, 



VG, '- vg, ■ 

The moment of the whole pyramid in reference to its base is 



6^1 (h + x) h + X _ 1 



h' g;- 



3 • ■ 4 ~ 12 [VG]- VG)'' 

and that of the part of pyramid, that is wanting, is 

h'VG^' 1 



%-'(»+!, 



■('-!) = I 



h' G^ 



7=^ -V ^ 



3 ^G,- VG, "^ 12 • {VG^ - VG.f 
hence the moment of the truncated pyramid is 
h' ^ . 

h' {G,' -4 G, ^G~G, 4- 3 a/) 



%' 



12 {G, - 2 VG, G,-\- G,) ~ 12 • ^^' + ^ ^^1 ^2 + 3 G,). 
Now the contents of the truncated pyramid are 

h 



V= {G, + VG, G, + G,) 



3' 



and therefore the distance of the centre of gravity S from the 
base is 



234 



GENERAL PRINCIPLES OF MECHANICS. 



[§13t 



y = 



6^1 + 2 VG, G, +d G, 



h 
4* 



G, + VG, G, + G, 
The distance 8, S of this point from the plane K L, passing 
through the middle of the body parallel to its base and dividing its 
height into two equal parts, is 



yi 



-^^y- l^(G^r+'^G^G.-¥G,)-(G,+ ^VG ^ G,+ 3 6^,)] h 

1 ex ij — , — '— — 



G, 4- VG^ G, + G, 

Gr - G, \ h 

4* 



^G, ■\-^/G, G,+ G, 
If the radii of the bases of 2^ frustum of com are Tx and 7*2, or 
Gi = TT Ti and G^ = -n r^, we have 



2^ = 



y^ 



r^ + 2 n r^ + 3 n 

Tx + /"i rs + T^ 



-7 and 
4 



Tx^ - r,' 



h 
4' 



rx'^ 4- r, ^2 + r^^ 

Example.— The centre of gravity of a truncated cone whose altitude 

is A = 20 inches and whose radii are r = 13 inches and r^ = 8 inches lies, 

as is always the case, in the line joining the centres of the bases, and at a 

distance 

.,-20 13' + 2. 1 2.8 + 3.8'' 5.528 2640 

y — -^ ' 102 , io — o . 02 = —K?rr~ — TT^;^ = 8,684 inches 



304 



304 



Fig. 157. 



12^ + 12 . 8 + 8' 
from the greater base. 

' "ti § 121. An oMish, i.e., a body A C Q, Fig. 157, bounded 
by two dissimilar rectangular bases and by four trapezoids, can 

be decomposed into a parallelopipedon 
A F R P, into two triangular prisms 
U R E Q Sind G KR and into a 
four-sided pyramid HK R. By the aid 
of the moments of these component 
parts we can find the centre of gravity 
of the whole body. 

It is easy to see that the right line 
joining the middle of one base to that 
of the other is a line of gravity of the 
body ; we have, therefore, but the distance of the centre of gravity 
from one of the bases to determine. Let us denote the length 
B C and the width ^ ^ of one base by Ix and hx, and the length 
Q R and the width P Q of the other base by Z2 and hc^, and the 
height of the body or the distance of the bases apart by h. The 




§121.] CENTRE OF GRAVITY. 235 

contents of the parallelopipedon are then = Js h Ti, and its moment 
is ^2 ?2 ^ • o = ^ bi h h\ The contents of the two triangular 
prisms are 

= ( [5, - h] h + Pi - y h) ^, 

and their moments are 

= {[h-b,]k+ [?i - y ^'2)^.3, 

and finally the contents of the pyramid are 

= (^1 — h) (?i — h) o> 
and its moment is 

= (h — in) (li — k) ^' J. 
From the above we deduce the volume of the whole body 

= (2 bill + 2 bi ?2 + §1 4 + h b^ 77, its moment 

o 

Vy =(fibih+2bxk+2 k h— 4.bjc,+ bi h + bi h— h h — Z, b^) . j^ 

= (3 bi ?2 i- bill -h bi ?2 + bi li) =^, 

and the distance of its centre of gravity S from the base bi li 
_ bill + S bcj Icj + bilcj + bi li h 
y ~ 2bili + 2b,h-\-bik + bTli ' 2' 
We can also put (see the " Planimetrie und Stereometrie " of 
C. Koppe) 

T^ _ bi + ^2 ?! + ?2 T bi — ^2 li — h \ 
^ ~ ~~2~~ ' 2 '^^ ~~2~ '2 • 3* 
The distance yi of the centre of gravity from the cross section 
through the middle is determined by the formula 

_ h bi Ix ' — bi l<i , 

y'~2~y- 3 {bi + ^^2) {k + k) + (^i - bi) ft -4) 

Remark. — This formula is also applicable to bodies with elliptical 
bases. If the semi-axes of one base are a^ and 5^ and those of the other 
ttg and &2, the volume of such a body is 

TT h 

F = -g- (3 «! 61 + 3 ag Jg + «! Jg + ttg \\ 

and the distance of its centre of gravity from the base tt a^ 5^ is 

«i &i + 3 «3 &g + a^ &2 + a^h^ h 
*^ ^ 2 a^ &i + 2 ^2 &2 + «! Sg + a2 ^1 * 2 * 



236 



GENERAL PRINCIPLES OF MECHANICS. 



[§ n^. 



Example. — If the embankment A C Q, Fig. 158, for a dam is 20 feet 
high, 250 feet long and 40 wide at the bottom, and 400 feet long and 15 

Fig. 158. 

B 
•0 




feet wide on top, what is the distance of its centre of gravity from its base ? 
Here &i = 40, l^ = 250, Jg = 15, l^ = 400, and h = 20, and consequently 
the distance is 

40 . 250 + 3 . 15 . 400 + 40 . 400 + 15 . 250 20 



y 



2.40 

4775 
5175 ■ 



. 250 + 2 . 15 . 400 + 40 . 400 + 15 . 250 * 2 

1910 
10 = -^ = 9,227 feet. 




§ 122. If the circular sector A CD, Fig. 159, is revolved about 
its radius C D, sl spherical sector A C B is generated, the centre 

of gravity of which can be determined in 
the following manner. We can consider 
this body as the aggregate of an infinite 
number of infinitely thin pyramids, whose 
common apex is the centre C and whose 
bases form the spherical zone AD B. The 
centres of gravity of each of these pyramids 
are situated at a distance equal to | of the 
radius C D of the sphere from its centre 
C, and they form a second spherical zone 
Ai Dx Bx, whose radius C D^ = ^ CD. 
The centre of gravity of this curved surface is also that of the 
spherical sector ; for the weights of the elementary pyramids are 
equally distributed over this surface, which is therefore every- 
where equally heavy. 

If we put the radius G A — C D — r and the altitude D M of 
the exterior zone = h, we have for the interior zone C D^ = | r* 
and M, D, = | h, and consequently (§ 116) 8 D,=: ^^ M,D, = I h, 
and the distance of the centre of gravity of the spherical sector 
from the centre C is 

CS= CD, - SD, = |r- |A-:|(r-|). 

For a hemisphere r = h, and therefore the distance of its centre 
of gravity S from the centre C is 



§ 128, 134.] 



CENTRE OF GRAVITY. 



237 



cs 



7z = i r. 



Fig. 160. 
D 



E^-Ei= 



— iF 



§ 123. A¥e obtain the centre of gravity yS' of a spherical seg- 
ment A B D, Fig. 160, by putting 
the moment of the segment equal 
to that of the spherical sector 
A D B G less that of the cone 
ABC. Denoting again the radius 
C D of the sphere by r and the 
altitude D M by h, we have the 
moment of the sector 
= I 77 r* /i . I (2 r - /i) = J ^ rVi (2 r - li), 
and that of the cone 

= \7t1i (2 r - h) . (r - h) . ^ {r - 7i) = {n 7i {2 r - h) (r - Ji)'; 
hence the moment of the segment is 

Vi/ = J TT A (2 r - h) {r' - [r - If]') = 
The contents of the segment are 

V = in li' (3 r - h), 
and consequently the required distance is 




TT ii' (2 r - ny. 



cs 



y = - — 



(2 r - 7iY 



n If [3 r - h) 



— 3 
~ 4 



(2 r - 7iy 
3r - 7i 



If we put again 7i — r, the segment becomes a hemisphere, and, 
as before, we have C S = | r. 

This formula is also true for the segment A^ D B^ of a spheroid 
generated by the revolution of the arc i> ^i of an ellipse about its 
major axis C i) = r ; for if we cut the two segments by means of 
planes parallel to the base A B into thin slices, the ratio of the 

corresponding slices is constant and = t L = ^ = —^, when 

MA' C E' ^ 
b denotes the smaller semi-axis of the ellipse. We must multiply 

not onlv the volume, but also the moment of the spherical segment 
by — ^ to obtain the volume and moment of the segment of the 



moment 

volume 
(2 r - 7i) 



is not changed. 



in which r de- 



spheroid, and therefore the quotient C 8 

In general we have G S = y — I „ 

^ ^ 6 r — h 

notes that semi-axis about which the ellipse is revolved, when gen- 
erating the spheroid. 

H,^§ 124. Application of Simpson's Rule. — In order to find 
the centre of gravity of an irregular body A B C D, Fig. 161, we 



ms 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 124. 



Fig. 161. 



divide it, by means of planes equally distant from each other, into 
thin slices and determine the area of the cross sections thus ob- 
tained and their moments in reference to the first parallel plane 

A B, which serves as base, and we then 
combine the latter by means of Simpson's 
rule. 

If the areas of the cross-sections are 
Fq, Fly F^, F-i, F^ and the total height or 
distance M N between the two parallel 
planes farthest apart = % we have, ac- 
cording to Simpson's rule, the volume of 
the body 

F= (i^o + 4i^, + 2i^, + ^F, + -F4) ^. 

Multiplying in this formula each surface by its distance from 
its base we obtain the moment of the body, viz., 




F^ = (0.i^o + 1.4i^i + 2.2i^2 + 3.4i^3 + 4i^4) 



4*12' 



and dividing the last equation by the first we obtain the required 
distance of the centre of gravity S 

(0 . i^o + 1 . 4i^i + 2 . 2i^a + 3 . 4i^3 + 4i^4) h 

4* 



M8 = y = 



y = 



F, + 4:F, +2i^2 + 4i^3 + i^4 

If the number of slices = 6, we have 

0.i^o + 1.4i^i4-2.2i^2 + 3.4i^3+4.2i^4 + 5.4i^5+6i<; 



h 



Fo +4 Fi + 2F, + 4i^3 +2i^4+4i^5 + i^6 ' 6* 

It is easy to see how this formula varies, when the number of 
slices is changed. The rule, however, requires, that the number of 
slices shall be an even one, or the number of surfaces an uneven one. 
In many cases we need determine but one distance, as a line of 
gravity is also known. Solids of rotation formed upon the turn- 
ing lathe are very common examples of such bodies. Their axis 
of rotation is a line of gravity. 

This formula is also applicable to the* determination of the 
centre of gravity of a surface, in which case the 
cross sections Fo, Fi, F^, etc., become lines. 

Example 1. For the parabolic conoid A B G, Fig. 
163, formed by the revolution of a portion A B M of 
a parabola about its axis A if, we obtain, when we make 
but one section i>i\r^ through the middle, the following. 

Let the altitude A M = h, the radius B M = r, 

AN — N M = ~ and consequently the radius D N 
= r Vf. Tlie area of the section through A\qFq = 0, 



Fig. 162. 




§ 135.] 



CENTRE OF GRAVITY. 



239 



Trr" 



that through JV F^ = tt B N^ = -^ and that through if, i^^g = tt r\ 

Hence it follows that the volume of this body is 
7) 7) 

V=q{0 + 4.F^ + F^) = -^i2nr' + nr') = ^n r'h = ^F^7i, 

and that its moment is 

Vy = ^il.2nr' + 2nr') = ^7rr'h' = iF^ h\ 

Consequently the distance of the centre of gravity S from the vertex is 



Fig. 163. 




3 







Example 2. The mean half widths of the vessel A B G B, Fig. 164, 
are r^ = 1 inch, r^ = 1,1 inches, 7*3 = 0,9 inches, ^'g = 0,7 inches, and r^ 
= 0,4 inches, and its height if iV= 3,5 inches; required the centre of 
gravity of the space within it. The cross sections are F^ = 1 tt, 
F^ = 1,31 TT, F^ = 0,81 7T, F^ = 0,49 n and F^ = 0,16 tt, and therefore 
the distance of its centre of gravity from the horizontal plane J. ^ is 

_ . 1 77 4- 1 . 4 . 1,21 TT + 2 . 2 . 0,81 TT + 3 . 4 . 0,49 tt + 4 . 0,16 . tt 2,5 



MS 



1 TT + 4 . 1,21 TT + 2 . 0,81 TT + 4 . 0,49 tt + 0,16 . TC 

14,60 2,5 _ 36,50 
- - -^ - 



9,58 



}8,32 



0,9502 inches. 



2,5 



The vacant space in the vessel is F= 9,58 rr . -f- 



6,270 cubic inches. 



(§ 125.) Determination of the Centre of Gravity of Sur- 
faces and Solids of Rotation. — The centre of gravity of curved 
Fig. 165. surfaces and of bodies with curved sur- 

faces can be determined generally by the 
aid of the calculus. In practice, solids 
and surfaces of rotation occur most fre- 
quently, and we will therefore here treat 
only of the determination of the centre 
of gravity of these forms. If the plane 
curve A P, Fig. 165, revolves about its 
axis A C, it describes a so-called surface 
of rotation A P P^; and if the surface 
A P M bounded by the curve ^ P an 1 




^^0 GENERAL PRINCIPLES OF MECHANICS. [§ 125. 

its co-ordinates A M and M P h revolved about the same axis a 
solid of rotation bounded by a circular surface P M P^ and by a 
surface of rotation ^ P Pi is produced. 

If we denote the abscissa A if by x^ the corresponding ordinate 
by y and the corresponding arc A P by s, and also the element 
M N ~ P R oi the abscissa by d x, the element Q R oi the ordi- 
nate by d y and the element P § of the curve by d s, we have the 
area of the belt-shaped element P Q QxP\ generated by the revo- 
lution of d s, when we put the surface of rotation A P P^ = 0, 

dO=^27T.PM,PQ = 27Tyds, 

and, on the contrary, the contents of the element of the solid of ro- 
tation A P Pi = V, limited by this element of the surface, are 

d V= 7T P^' .M]Sr=7Ty'dx. 
Since the distance of both elements from a plane passing 
through A at right angles to the axis A Cis equal to the abscissa 
X, the moment of tZ is 

xdO=2Trxyds, 
and that ofd F is 

X d V = TT x y^ d X. 
Now since 

= f2nyds = 27Tfyds and 
V = f ny^ d X = n / y"^ d X, 

and since according to the above formulas the moment of is 

f^TTxyds — 2 n / X y d s, 
and that of V is 

fiTxy^dx = 7T/xy^dx, 

it follows, that the distance A S = yof the centre of gravity Strom 
the origin A is 

1) for surfaces of rotation 

_ 27rfxyds__ fxyd s 
~ 2 7T f y d s ~~ S yd i 
and, on the contrary, 

2) for sohds of rotation, 

_!: f X y"" dx _ fxy^dx 

~ -n f y"^ dx ~ f y^ dx* 

E.G., for a spherical zone whose radius (7 § = r we have, since 

P Q G Q ds r , , 

^r— = ^;— ^i.E. -Y- — — Qxy d s — rdx, 
P R Q N dx y ^ 



§ 126.] 



CENTRE OP GRAVITY. 

/ X r d X _ fxdx _ £^ 
frdx "" fdx ~ X 



241 



^ X 



^AM. 



(Compare § 116.) 

For a segment of a sphere, on the contrary, we have, since we 
can put y"" = 2 r X — x"", 

f {2rx — x'')xdx _ f%rx^dx — fx^dx 



A S = u = 



f {%rx — x')dx f%rxdx — fx'Ux 



^rx^ 



- ix' _ {%r-ix) X _ ( Sr-3x 



rx 



1 ^3 

3 -^ 



r 



1 /y* 

3 U. 



(8 r — 3x\x 
dr-x ) ? 



and consequently 



CS=r 



„ (2 7- - xy 

^ or — X 



(Compare § 123.) 



§ 126. Properties of Guldinus. — An interesting and often 
yery useful application of the theory of the centre of gravity is 
the properties of Guldinus (Fr. methode centrobarique, Ger. die 
Guldinische Eegel). According to these the contents of a solid 
of rotation (or the area of a surface of rotation) is equal to the 
■product of the generating surface (or generating line) and the 
space described hy its centre of gravity ivhile generating the tody 
(or surface). The correctness of this rule can be proved as follows : 

K a plane surface A B D, Fig. 166, is revolved about an axis 
XX, every element Fi, F^, etc. of it describes a ring; if the dis- 
tances of these elements Fi, F^, etc. from the axis of rotation 

XX are F^ K^, F^^ K^, etc. = ri, ra, etc., 
and if the angle of rotation \& F K F^ 
=^ S C Si = a° or the arc corresponding 
to the radius 1, == a, the arc- shaped 
paths described by the elements are 
ri a, r<i a, etc. The spaces described by 
the elements F^, F^, etc., can be re- 
garded as curved prisms whose alti- 
tudes are r^ a, r^ a, etc., their contents 
are therefore F^ ri a, F^ r^ a, etc., and 
consequently the volume of the whole body A B D Di B^ Ai is 

V= F^r, a + F^r^a + = (Fir, + F^r.^ + ...) a. 

16 



Fm. 166. 




242 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 126. 



If ?/ = CS is the distance of tlie centre of gravity S of the gen- 
erating surface from the axis of rotation, we have 

{F, + F. -h. . . .) y — F, r, + F2 r^ + . . ., 
and consequently the volume of the whole body 

V= {F, + F, + ...)ya. 

But i^i + i^2 H- ..• is the area of the surface F, and y ais the arc 
S 81 = w described by the centre of gravity; hence it follows 
that V =' F tv, which is what was to be proved. 

This formula is also applicable to the case of the rotation of a 
line, since the latter can be considered as a surface of infinitely 
small width. In this instance we have F = I tv, i.e. the surface 
of rotation is the product of the generating iine (/) and the space 
{iv) described by its centre of gravity. 

Example 1. If the semi-axes of the elHptical cross section ABED, 
Fig, 167, of a half ring Sire C A = a and C B = d, and if the distance C 31 

of its centre G from the axis X X = r, the elliptical generating surface 
will be ^ = 7T ad, and the space described by its centre of gravity (C) 
will he w = 7T r. Hence the volume of this half ring is F = n^ ad r, and 
that of the whole ring is V^ ~ 2 V = 2 tt^ ad r. 

If the dimensions are a = 5 inches, 5=3 inches and r = 6 inches, the 
volume of one-quarter of the ring is 

I- . TT^ . 5 . 3 . 6 = 9,8696 .5.9= 444,132 cubic inches. 





Example 2. The volume of a ring with the semi-circular cross section 
ABB, Fig. 168, is, when G A-= G B = a denotes the radius of this cross 
section and M G =^ r that of the hollow space, 



V = 



K a' 

~2~ 



. 2 TT (r -f- — I = 7r «^ (tt r + f a). 



Example 3. If the segment of a circle ABB, Fig. 169, revolves about 
the diameter ^i^ parallel to its chord A B, it describes a sphere A B^B 
with a cylindrical hole A B B^ A^in it. If A is the area of the segment 



126] 



CENTRE OF GRAVITY 



243 



and s the length of its chordJ^ B = A^ i?j, we have (§ 114) for the distance 
of its centre of gravity 8 from the centre C 

C8=y = 



12 A' 



and consequently the volume of the sphere with the cylindrical hole is 



V=2nyA = 27T 



TC S' 



12 6 ■ 

For a complete sphere we have the chord or height of the hole equal to 
the diameter d of the sphere, and consequently its volume 

TTd:' 



V = 



6 ' 



as we know. 



Example 4. We are required to find the area of the surface and the 
contents of the cupola ABB, Fig. 170, of a cloistered arch, when the half 





width M A = 3f B = a and the altitude M B = li are given. From the 
two given dimensions we obtain the radius G A = G B oi the generating 
circle 



T = 



2 a 



The central angle J. C Z) = a is given by the formula 

li 



sm. o = -. 
r 



The centre of gravity S of an arc i) ^ -Dj = 2 ^ -D is determined by 
the distances 



G S = r 



chord MB r sin. a 



and G M = rcos. a ; 



arc A B a 

consequently the distance of the centre of gravity 8 from the axis if i) is 

,^ _ r sin. a /sin. a \ 

M if = — r COS. a = r { COS. a |, 

a \ a r 

and the space described by the centre of gravity in describing the surface 

AB B\% 



w 



(sin. a \ 

= 2 TT r I COS. a\. 



244 GENERAL PRINCIPLES OF MECHANICS. [§127. 

The generatrix I) A D^ is 2 r a, consequently its half is J. i) = r a, and 
the surface of rotation ABB generated by the latter is 

(sin. a \ 
COS. a \ =2 Tvr^ (sin. a — a cos, a). 

Very often we have a° = 60°, or 

a = -, sin. a = |- V3 and cos. a = |- ; 

hence the required area is 

= TT r^ /V3 - ^) = 3,1515 . r\ 

The distance of the centre of gravity of the segment B AB^ = J. = r* 
(a — ^ sin. 2 a) from the centre C is 

_ {2.MBy _ 2 rUin.^a 
~ WA ~ 3 * A ' 

and, therefore, its distance from the axis is 

2 v"^ sin ^ d 
M8= 08- GM= - -^ rcos.a, 

and the space described by this centre of gravity in one revolution around 
M B \s 

2 TC T , 2 7T t'^ . 

w = —z — (f r^ sin.^ a — A cos. a) = — ^ — [f sin.^ a — (a — |- sin. 2 a) cos. a]. 

The volume of the body generated by the revolution of the segment 
B A B^ is found by multiplying this space by A, and the volume of the 
cupola by dividing the last product by two. The latter volume is 

V= TT r^ [f sin.^ a — (a — ^ sin. 2 a) cos. a] 
E.a, if a° = 60°, we have 

e = -, sin. a —^ V3, sin. 2 a = ^ V3, cos. a = ^, and therefore 



F = TT 7"M| V3 — ^) = 0.3956 . r\ 



§ 127. The properties of Guldinus are also applicable to bodies 
formed by the motion of the centre of gravity of the generating^ 
surface along any curve, as long as the surface remains at right- 
angles to the curve ; for every curve can be regarded as composed 
of an infinite number of infinitely small arcs of circles. The vol- 
ume of the body is here also equal to the product of the generating 
surface and of the space described by its centre of gravity. The 
properties can also be made use of, when the generating surface in 
moving forwards is always at right angles to the projection of the 
path of its centre of gravity upon any plane. In this case the 
generating surface is to be multiplied not by the space described^ 
but by its projection. 



128.] 



CENTRE OF GRAVITY. 



245 



Fig. 171. 



Hence, for example, the volume of one turn of the thread 

AUK, Fig 171, of a screw is de- 
termined by the product of its cross 
section A B D B hj the circum- 
ference of the circle, whose radius 
is the distance M S of the centre 
of gravity S of the surface A B D E 
from the axis C M oi the screw. 

In many cases we can combine 
the use of the properties of Guldi- 
nus with that of Simpson's rule. 
E.G., to find the contents of the 
curved embankment A^DqB^ i)o A^y 
Fig. 172, we need only know the central angles So C S.2 = 2 So CSi 
= 2 SiC S^= (3, the cross sections Aq Do — Fo,A^D^ — F^, A^ A 





= F^ and the distances C So = To, C Si = r^ and C S^ = ^s of the 
centres of gravity So, Si and S^ of these cross sections from the cen- 
tral axis C ^. The volume V of the body is determined by the 
formula 

( Fq ro-i-4.Firi + F, r, \ _ ^ ( Fp n + 4 .F, r, + F, rA 
^ ^ \ 6 / ~ 180*^ V 6 / 



= 0,01745 /3° / 



„ /Fo n -^ 4: FiTi + F, r. 



6 



If the radii Tq, Vi and ra are equal to each other, or if they differ 
but little, we can put ro = r, = r^ = r and therefore 



V = 0,01745 (i' ri 



Fq + 4.Fi + F, 
6 



)■ 



§ 128. The following is another application of the theory of 
the centre of gravity, which is closely allied to the foregoing. 



246 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 138. 



We can assume that every obliquely truncated prismatic body 
A B K L, Fig. 173, is composed of infinitely tliin prisms, such as 

Fi Gx. If Gi, G^j, etc., are the bases and 
hi ^2, etc., the altitudes of these prismatic 
elements, we have the contents 

Gi hi, G<2 h^, etc. 
and consequently the volume of the 
whole obliquely truncated prism 

V = Gi hi + 6^2/^2 + 

Now an element F^ of the oblique 
section ^ Z is to the element Gi of the 
base A B = 6^ as the whole oblique sur- 
face F is to the base G : hence we have 




G 



G 



G\ — y Fly ^2 = ^ ^2, etc., and 
F = I (i^, ^, + F, A2 + . . .). 



Finally, since Fih^ + F^h^^ + . 
whole oblique section, we can put 

V= ^. Fh 
F 



is the moment Fh of the 



Gh, 



I.E., the volume of an oUiquely truncated jorism is equal to the volume 
of a complete prism, which stands on the same base and whose alti- 
tude is equal to the distance 8 of the centre of gravity 8 of the 
ohlique section from the hase. 

The distance of the centre of gravity of the oblique section of a 
right triangular pi'ism, which is truncated obliquely, from the 
base is 

hi 4- 7^2 + A3 



h = 



3 



and consequently the volume of this prism is 

V= Gh= G (hJiJL^JlM, 



§129.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



247 



CHAPTER III. 

EQUILIBRIUM OF BODIES RIGIDLY FASTENED AND SUPPORTED. 

§ 129. Method of Fastening. — The propositions relative to 
the equiUbrium of rigid systems of forces, demonstrated in the first 
chapter of this section, are apphcable to solid bodies subjected to 
the action of forces, when we consider the weight of the lody as a 
force apiMed at the centre of gravity and acting vertically down- 
ivards. 

Bodies, which are held in equilibrium by forces, are capable of 
moving freely, i.e., they can obey the influence of the forces, or 
they are in one or more points rigidly fastened, or they are sup- 
ported by other bodies. 

If a point C, Fig. 174, of a solid body is rigidly fastened, any 

Fig. 174. 




other point P of the body, when put in motion, will describe a path, 
which lies upon the surface of a sphere, whose centre is the fixed 
point C and whose radius is the distance C F of the other point 
from C. If, on the contrary, we fasten a body in two points C 
and D, the paths described by all other points in consequence of 
any possible motion would be circles ; for the path of each point is 
the intersection F Q of two spherical surfaces described from 
the two fixed points. 

The planes of these circles are parallel to each other and per- 
pendicular to the straight line joining the two fixed points. The 
points upon the latter line remain immovable ; the body, therefore. 



248 GENERAL PRINCIPLES OF MECHANICS. [§ 13Q. 

revolves around this line C D, which is called, for this reason, the 
axis of rotation or revolution of the body. 

The planes perpendicular to this axis, and in which the different 
points revolve, are called the ]jilanes of rotation or revolution of the 
body. We obtain the radius M P of the circle P Qhj letting 
fall a perpendicular upon the axis of revolution C D, The greater 
this perpendicular is, the greater is the circle, in which the point 
revolves. 

If three points of a body, not in the same straight line, are firmly 
fastened, then the body does not move in any direction, since 
the three spherical surfaces, in which the body must move, cut each 
other only in a point. 

130. Equilibrium of Supported Bodies. — Every force pass- 
ing through the fixed point of a body, e.g., through the centre of a 
ball and socket joint, is counteracted by the support of the body, 
and has, therefore, no influence upon the state of equilibrium of 
the body. In like manner, if a body is supported in two points or 
bearings, every force whose direction cuts the axis passing through 
these fixed points is counteracted by the supports, without pro- 
ducing any other effect on the body. A couple would also be 
counteracted by the supports of a body, if the plane of the couple 
contains the axis of revolution passing through these points, or is 
parallel to the same. Every other couple (P, — P), Fig. 175, 
produces, on the contrary, a revolution of the body A C B about 
the axis of revolution C, if it is not balanced by another couple 
(see § 95 and § 97). If the couple retains its direction during the 
rotation, its lever arm and consequently its moment is variable, and 
both become = 0, when the body occupies a certain position. If a 

body A C B, Fig. 175, is rigidly fast- 

FiG. 175. ened at C, and if the direction of the 

^^^ ^^ force forms the angle B A P = a with 

4^^^- .^y^ the line A B passing through the 

^k ..•••''/ * two points of application, a rotation 

l^k' / A C Ai = 13 = ISO" — a is necessary 

yf^^^ : to annul the moment of the couple 

^'^^^^^^ (P, — P) ; the same is also true of a 

y^^ ^^^®^ body rigidly fastened in an axis and 

-^ I^^^M acted upon by a couple, whose plane 

is perpendicular to this axis. 
If a body A B, Fig. 176, rigidly fastened at (7, is acted on by a 



§ 131.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



249 



force P, whose direction does not pass through C, we can, by the 
addition of two opposite forces P and — P, decompose this force 
into a couple (P, — P) and a force + P, applied in C and coun- 
teracted by the point of support. The rela- 
tions are the same, when the axis of a body is 
rigidly fastened and a force acts upon it in a 
plane of revolution. Here, however, the force 
+ P is divided between the two points of sup- 
'■^1 port. If a is the distance G A of the point of 




application A of the force from the axis C and a 
the angle A C A^, formed by the line C A with 
the direction of the force, we have the moment 
of the couple (P, — P), which tends to turn the 
body, M = Pa sin. a. If the direction of the 
force P remains unchanged during the rotation, 
M changes with a and is a maximum for a = 90° 
and for a = 0° or 180° it is = 0. The work done by the force 
P or by the couple (P, — P) during the rotation of the body is 

A = P . WA, = Pa{l - COS. a). 

131, Stability of a Suspended Body. — If the force acting 
upon a body, supported at one point or in a line, consists only of 
its weight, the conditions of equilibrium require, that the centre of 
gravity shall be supported, i.e., that the vertical line of gravity 
shall pass through the point of support. 

If the centre of gravity coincides with the point of support, we 
have a case of indifferent equilibrium (Fr. equilibre indifferent, Ger. 
indifferentes Gleichgewicht) ; for the body remains in equilibrium. 



Fig. 177. 



Fig. 178. 





no matter how we may turn it. If, on the contrary, the body is 



250 



GENERAL PRINCIPLES OF MECHANICS. 



[% 132. 



rigidly fastened or supported at a point C, lying above the centre 
of gravity S, the body is in stable equilibrium (Fr. stable, Ger. sich- 
eres or stabiles) ; for, if we bring the body into another position, one 
of the components JV of the weight S causes the body to return to 
its original position, and the other component P is counteracted by 
the fixed point C. If finally the body A B, Fig. 178, is fastened 
at a point G, which lies below the centre of gravity, the body 
is in unstalle equilibrium (Fr. eq. instable, Ger. unsicheres or 
labiles Gleichgewicht) ; for if we move the centre of gravity out of 
the vertical line passing through (7, the weight G is resolved into 
two components, one N of which, instead of tending to bring the 
body back to its original position, moves it more and more from it, 
until the centre of gravity comes vertically below the point of 
support. 

The circumstances are the same, when a body is supported in 
two points or in an axis ; it is either in indifferent, stable or unstable 
equilibrium as the centre of gravity coincides with, or is vertically 
below or above the point of support. If a body is supported at a 
point or in a horizontal axis, the moment with which the body seeks 
to return to its position of stable equilibrium is if = G a sin. a, 
in which formula G denotes the weight, a the distance G 8x of the 
centre of gravity 8x from the axis G and a the angle of revolution 
S G Si. The work done is A = G a (1 ~ cos. a). 

§ 132. Pressure upon the Points of Support of a Body. 
— When a body GAD, Fig. 179, supported in two points G and 

Fig. 179. 




Df is acted upon by a system of forces, in order to determine the 
conditions of its equilibrium we refer (according to § 97) the 



§ 132.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



251 



whole system to two forces, the direction of one of which is parallel 
to the axis, while that of the other lies in a plane normal to this 

line. Let EN— N, Fig. 180, be the force parallel to the^axis XX 
passing through the points of support C and D and A P ^^ F the 

other force, whose direction lies in a plane Y Z Y perpendicular to 

XX. We can resolve the first force into a force + N, tending to 
displace the axis in its own direction, and a couple ( iV, — N), 
which is transmitted to the points of support in the shape of an- 
other couple (iV„ — iVi), the components of which are 

Nx=~ N and - N^ = — j N, 

(I denoting the distance E of the parallel force N from the axis 
C D and I the distance C /> of the two points of support from 
each other. 




In like manner we decompose the force P into a force + P and 
a couple (P, — P), and the former again into its components P, 
and Ps? the first applied in C and the second in D. Designating 
the distances C and D oi the points of application from the 
two points of support G and D by l^ and Z^, we have 



P, = 1l p and P, 






and it is now easy, by employing the parallelogram of forces, to find 
the resultant 8x of the forces JV, and Pj at C, and also the resultant 
Si of the forces — iVi and P2 at D. 

If we put the angle Y {+ P) formed by the plane N X 
with the direction of the force P or + J^ = a, we have also the 



252 GENERAL PRINCIPLES OF MECHANICS [^ 133. 

angle Nx C Px — a and N^ D Pc^ = 180" — a, and consequently the 
resulting pressures in C and D are 

8, = VN,' + P;' + %N,P, COS. a 
and S, = VJV,' + P,' -%N,P, cos. a. 

If, finally, a denotes the perpendicular L to the direction of 
the force, the moment of the couple (P, — P), which tends to turn 
the body, is M = P a. If the body is in a state of equihbrium, a 
must naturally be = 0, and therefore P must pass through the 
axis CD. 

Example. — Let the entire system of forces acting on a body rigidly 

supported in the axis X Xbe reduced to tbe normal force P = 36 pounds, 
and the parallel force J^ = 20 pounds ; let the distance of the latter force 
from the axis he E = d = 1^ feet, and the distance C D between the 
two points of support be ? = 4 feet ; required the pressure upon the axis 
or on the fixed points G and D supposing that the direction of the force P 
forms an angle a = 65° with the plane X Y, and that its point of applica- 
tion is at a distance G = l^ =: 1 foot from the point G. 

The force iV = 20 produces in the axis in its own direction a thrust 
i\r = 20 pounds and also the forces 

d 15 

If^ = J- N =~- .20 = 1,5 pounds and — J!^^ = — 7,5 pounds, 

which are counteracted by the supports (7 and JD. The force P gives rise to 
the forces 

Pj = y P= -^— . 36 = 27 pounds and P^ = -jP = l. 36= 9pounds. 

Combining the latter with the former force, we obtain the resultants 



S^ = VT,5^ + 27^ + 2 . 7,5 . 27 . cos. 65° = V 56,25 + 729 + 171,160 

= V 956,410 = 30,926 pounds, and 
Sz = V 7,5'^ + 9=^ — 2 . 7,5. 9. cos. 65° = V 56,25 + 81 — 57,054 
,, = V 80,196 = 8,955 pounds. 

§ 133. If a body G B D, Fig. 181, firmly supported in two 
points C and D, is acted upon by a single force P, whose direction 
forms an angle P A B — (3 with the plane of rotation Y Z, we 
can decompose this force into the components 

jTP =z P == B cos. P and 



A N=N= Bsin.l3, 
the first of which acts in the plane of rotation and the second 
parallel to the axis, and we can treat these forces in exactly the 
same manner as the resultants P and N of the system of forces in 



§133.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED 



253 



the last paragraph. Here the force which the axis must counter- 
act in its own direction is N = R sin. 13, and the components of 




the couple (JVi, — iVi), which act in Cand D in opposite directions 
and at right angles to D, are 

d 



d d 

N^=jN= J R sin. 13 and — iVl = 



I 



R sin. (3, 



I denoting the distance O D of the two points of support C and D 
from each other and d the distance A of the point of application 
A of the force R from the point on the axis. 

In like manner the force acting in at right angles to G D ia 
■^ P — R cos. (3 and its components in C are 

Px = jP — jR COS. (3, and in D 

P, = ^jP = ^jRcos.l3, 

li and k again denoting the distances C and D of the points C 
and D from the plane of rotation Y Z Y. 

Substituting the values of iV"„ P„ and Pg in the formulas 

8, = V N,' + Pi' -\-2N,P,cos.a 



8, = Vn7+P,' -%N,P, cos. a 

for the normal pressures in C and D, in which we designate by a 
the angle YAP formed by the component P with the plane 
A C D,yfQ obtain 



254 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 134. 



R 



Si= -r-V (d sin. ay + (?2 COS. ISy -\- 2 d k sin. /3 cos. (3 cos. a 

S^= y-^^^ ^*^* ^y '^ (^^ ^^^' ^y — ^^^ k'sv)i^ cos. 13 cos. a 
The moment of the remaining conple (P, — F) is 



Fig. 182. 




F.OB = Fa — Rd sin. a cos. 13. 

These formulas are applicable to the discussion of the stability 
of a body A, Fig. 182, revolving about an inclined axis CD. R 

is here the weight G of the body, 
t? the distance 8= S^ of its 
centre of gravity from the axis of 
rotation, a the angle S Si= SiL, 
which the centre of gravity has de- 
scribed in turning from its position 
of equilibrium S in the plane Y S K 
perpendicular to CD, and (3 the angle 
G Si F formed by the plane of revo- 
lution with the vertical line, or that 
formed by the axis of revolution CD 
with the horizontal line D H. 

The work done, when the body is 
brought back by its weight to its position of equi*librium and 
8i to 8, is 

A = G . K 8 COS. (3 — G d cos. (3 {1 — cos. a). 

§ X34. Equilibrium of Forces around an Axis. — The re- 
sultant F is produced by all the component forces, whose directions 
lie in one or more planes normal to the axis. But in this case 
(according to § 89) the statical moment P a is equal to the sum 
Pi ^1 4- Pa <^2 + • • • of the statical moments of the components, 
and, when the forces are in equilibrium, the arm a is -- ; for this 
force then passes through the axis itself, and consequently this sum 

Pi flJi H- Pg ^2 4- . . . = ; 

I.E., a body rigidly supported in an axis is in equilibrium, and 
therefore remains without turning, when the sum of the statical 
moments of all the forces in relation to this axis is = 0, cr when 
the sum of the moments of the forces acting in one direction of 



§135.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 255 

rotation is equal to the sum of the moments of those acting in the 
other. 

By the aid of the last formula any element of a balanced sys- 
tem of forces, such as a force or an arm, can be found, and any 
force of rotation reduced from one arm to another. 

If we wish to produce a state of equilibrium in a body movable 
about its axis, and whose moment of rotation is P a, we have only 
to apply a force of rotation § or a couple, the moment of which 
Qh = P a, the difference in the two cases being that by the addi- 
tion of the couple {Q, — Q) the pressure on the axis is not changed, 
while by that of a force Q a force + § is added to the pressure on 
the axis. If the fort^ Q or its lever arm b is given, we can calcu- 
late either 

, Pa ^ Pa 

In the latter case we call Q the force P reduced from the arm a 
to the arm &, and we can thus reduce the given force of rotation P 
to any arbitrary arm, or we can replace or balance it by another 
force acting with any arbitrary arm. 

We can also, by means of the formula 

^ X\ C?J ~n -t g a^ "T" ... 

^ = - -f , 

reduce a whole system of forces to one and the same arm. 

Example. — The forces P^ = 50 pounds and Pg == — 35 pounds act 

on a body movable about an axis with the arms a^ = 1^ feet and a^ = 

2^ feet ; required the force i^ which must act with an arm a^ = 4: feet, in 

order to produce equilibrium or to prevent motion about the axis. We 

have 

50 . 1,25 — 35 . 2,5 + 4 P., = 0, and 

P 87,5-62,5 __^ ' . 
> (. ^ ^s = —^ — -T — = 6,25 pounds. 

§ 135. The Lever. — A body movable about a fixed axis and 
acted on by forces is called a lever (Fr. levier, Ger. Hebel). If we 
imagine it imponderable, we have a mathematical lever; but if not, 
it is a material lever. 

We generally assume the forces of a lever to act in a plane at 
right angles to the axis and substitute for the axis a fixed point 
called the fulcrum (Fr. point d'appui, Ger. Euhe, Dreh, or Stiitz- 
punkt). The perpendiculars let fall from this point upon the di- 
rection of the forces are called (§ 89) the arms of the lever. If the 
directions of the forces of a lever are parallel, the arms of the lever 



256 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 136. 



form a single right line, and the lever is then called a straight 
lever (Fr. levier droit, Ger. geradliniger or gerader Hebel). The 
straight lever acted on by two forces only is one or two armed, ac- 
cording as the points of application of the forces lie upon the same 
or upon opposite sides of the fulcrum. We distinguish also levers 
of the first, second and third sort, calling the two-armed lever a 
lever of the first sort, the one-armed lever a lever of the seconc 
sort or of the third sort, according as the force (load), which acts 
vertically downwards, or that (power), which acts vertically up- 
wards, is nearest the fulcrum. 

§ 136. The theory of the equilibrium of the lever has been 
completely demonstrated in what precedes, and we have only to 
make special applications of it. 

For the two-armed lever A C B, Fig, 183, when the arm C A 
of the force P is denoted by a and that B of the other force ft 
which is generally called the load, by J, we have, according to the 
general theory P a= Qh, i.e. the moment of the force is equal to 



Fig. 183. 



A(^ 



~^i 



\ 



Fia. 184. 



«B 






A(^ 



-^ 



Q 



B 



the moment of the load, or also P : Q = h\a, I.E. the force is to the 
load as the arm of the latter is to the arm of the former. The 
pressure on the fulcrum is R = P + Q. 

For the one-armed lever ABO, Fig. 184 and B A C, Fig. 185, 
the relations between force (P) and load ( Q) are the same, but the 
direction of the power is opposite to that of the load, and therefore 
the pressure on the fulcrum is equal to the difference of the two ; in 
the first case we have 



E= Q — P, and in the second R = P — Q. 



§136.J EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 257 



If in the bent lever A C B the arms are C N= a and C 
I, Fig. 186, we have again P \ Q — b\a, but in this case the 

Fig. 185. Fig. 1*86. 



E 



Bc^ 



^ 



T5 



io 




pressure R on the fulcrum is the diagonal R of the parallelogram 
C Px R Qi, constructed with the force P, the load Q and with the 
angle Pi CQ^ — PDQ = a formed by their directions with each 
other. 

If G is the weight of the lever and C E = e, Fig. 187, the dis- 
tance of the fulcrum C from the vertical line 8 passing through 
the centre of gravity /S' of the lever, we must put P a db G e = Qh, 
and we must employ the plus sign of G. when the centre of gravity 
lies on the same side as the force P, and the minus sign, when 
upon that of the load Q. 

The theory of the lever is often applicable to tools and ma- 

FiG. 187. Fig. 188. 



-nE 





I>X p 

chinery. The knee lever A B C D, Fig. 188, which is sometimes 
cited as a peculiar sort of lever, is simply a bent lever. The arm, 
which is movable around an axis C, is acted upon by a force at its 
17 



258 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 136. 



end A, and acts by means of a rod B D, (which forms with the arm 
an acute angle A B D = C B E = a) upon the load, which is ap- 
plied at D. If a denotes the length of the arm G A and i the 
length of the arm C B, we have the lever arm of Q 



G E 


— 


h sin. a, 


whence 




Pa 


= 


Q i sin. 


a, or 




P = 


a 


Q sin. a 


and inversely 


n — 


a 


P 







The 



Fig. 189. 




h sin. a 

This lever is employed for pressing together materials. 

■ ci 
pressure increases directly with P and -, and inversely as sin. a. By 

diminishing the angle a this force Q can be arbitrarily increased. 

Example — 1) If the end Jl of a crowbar A C B, Fig. 189, be pressed 
down with a force P of 60 pounds, and if the arm C ^ of the power is 13 

times as great as the arm B 
of the load, then the latter, or 
rather the force Q developed in 
i?, is 12 times as great as P, and 
we have 

Q = 12 . 60 = 720 pounds. 

2) If a load Q, Fig. 190, hang- 
ing from a bar, be carried by 
two workmen, one of whom 
takes hold at A and the other 
^t P, we can determine how 
much weight each has to sus- 
tain. Let the load be Q — 120 
pounds, the weight of the 
rod be (r = 12 pounds, the 
distance A B oi the two work- 
men from each other be = 6 
feet, the distance of the load 
from one of them B he B G ^ 
2|- feet and the distance of the 
centre of gravity of the bar S 
from the same point he B S = 
3 J feet. If we regard B as the 
fulcrum, the force P^ at A must 
balance the load Q and G, and 
therefore we have 



^^Q 



Fig. 190. 




§ 137.] EQUILIBRIUM OF BODIES RlGIDLi FASTENED. 



259 



P^. BA= Q . B C + G . B S, i.e., 

6 Pi = 2,5 . 120 + 3,5 . 12 = 300 + 42 = 342, 



and therefore 



Pi 



342 



= 57 pounds. 



If, on the contrary, A be regarded as tiie fulcrum, we can put 
Pg . A B = Q . A C + G . A 8^ ov in numbers 
6 Pg = 3,5 . 120 + 2,5 . 12 = 420 + 30 = 450, . 
and the force exerted of the second workman is 

450 , ' 

Pg = — = 75 pounds. 

The sum of the forces, which act upwards, is therefore correctly 

P^ + Pg = 57 + 75 =132 pounds, 
or as great as the sum of those acting downwards 

Q + G^ = 120 + 12 = 132 pounds. 

3) The load upon a bent lever A G B, Fig. 191, weighmg 150 pounds, 
acts vertically downwards and is Q = 650 pounds, and its arm G B = 4: 

feet, and, on the contrary, the arm of the force 
P,GA = Q feet and that of the weight GB=1 foot : 
required the force P necessary to produce equili- 
brium and the pressure B on the bearings. We have 

G~A . P = GB. Q + (Te. G, i.e., 

6 P = 4 . 650 + 1 . 150 = 2750, 

and consequently 

2750 
P = — ^ = 458|^ pounds. 

The pressure on the bearings is composed of the 
vertical force Q. + G^ = 650 4- 150 = 800 pounds, 
and of the horizontal force P = 458|^ pounds, and 
consequently we have 



T<- 



Fig. 191. 



# 



13 



Q^ 



P 



'y'iQ + Gf + P' 
= VTSOOJM^ (458^)^ 
= V"850070 = 922 pounds. 

§137. More than two forces P and (J niay act on a lever; it 
also is not necessary that these forces act upon the lever in one and 
the same plane of rotation. If Q^, Q^, Q.^ are the loads on a lever 
A OBi,¥ig. 192, and b„ b,, b, their lever arms O B„ C B„ C B,, 
while the power acts with the lever arm C A = a, we have 

P a ^ Q,b, + Qrb, + Q,h\ 
and if the lever is straight, the pressure on the fulcrum is 

7? =3 P 4- e. + e. + ft- 
If the several forces of a lever act in different planes of rotation 



260 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 137. 



upon the leyer A G JD B^ B^, Fig. 193, the formula for the moment 
P a =^ Qiii + ^2 ^2 + • . • does not therefore change, but a differ- 
ent distribution of the total pressure B = P + Qi -h Q^ + Qt 

Fig. 193. Fig. 193. 



A. 


C B 


X. Ba Ba 


V 


> 
( 

< 




U 




upon the axis takes place between the two points of support or 
bearings C and D, If we denote by / the length of the axis 
C D oi. the leyer or the distance of the fulcrums from each other 
and by lo, l^, k, . . . the distances G 0, C 0], G 0^ of the planes of 
revolution from the fulcrum G, the pressures B^ and B^ on the. 
bearings at D and G are determined by the following formulas 



Bo 



Bi — B — B<i 



I 



and 



I 



If the forces acting upon a bent lever are not parallel, the ex- 
pression P a = Qi hi -{- Qc,b<i + ... remains unchanged, but the 

pressures in the axis reduced to the fulcrum, e,g., -^--, -—--, ~-^, act 

etc 

in different directions and cannot, therefore, be combined by simple 

addition, but, on the contrary, we must combine them in the same 

manner as several forces applied to a point and acting in the same 

plane (see §§ 79 and 80). 

Example. — The lever represented in Fig. 193 supports the loads Q^ = 
300 pounds and Q^ = 480, acting at the distances 0^ = l^ = 12 inches 
and (7 Og = Zg = 24 inches from the bearing (7 with the arms 0^ B ^ = 
l^ = 16 inches and Og B^ =d^ = 10 inches; required the force P, which, 
acting with the arm A = a = QO inches, is necessary to produce equili- 
brium, and the pressure on the bearings at G and Z), under the assumption, 
that the force acts at a distance C = l^ = 18 inches from the journal (7,. 
and that the length of the entire axis is G B = I = d2 inches. 

The force required is 

&, 300.16 + 480.10 30.16 + 480 



p_ Qx^l + Q2-2 

a 60 

pounds, and the pressures on the bearings are 



6 



= 80 + 80 = 160 



138.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 261 



Bq = 



160 . 18 + 300 . 12 + 480 . 24 
32 



=562,5 pounds and 



Fig. 194. 



S 



"ZS — z^ 



3 



lA 



i?j = i^ — jRg = 300 + 480 + 160 — 562,5 = 377,5 pounds. 

Remark.— The action of gravity on the lever can be employed with 

advantage to determine the centre of gravity S> and the weight G^ of a 

body A B, Fig. 194. We support the body 
first at a point C and then at a point G^ at a 
distance C C^ = d from the former, and each 
time we bring the body into equilibrium by a 
force acting at the distances C A = a and 
C^ A = a^ = a — d. If the force necessary 
^ in the first case be = P and in the second case 

= Pj, and if the weight of the body be G and 
the distance of its centre of gravity S from A he A B = x, yfe have 
P a = G (x — a) and P^ a^ — G (x — a^), whence 









^< 



G 



Pa — P^a^ 

Pa — II a^ 

a^ — ^1 



Fig. 195. 



§ 138. Pressure of Bodies upon one another. — The law 

deduced from experiment and announced in § 65 : " Action and 
reaction are equal to each other," is the basis of the whole mechan- 
ics of machines, and we must liere explain at greater length its 
meaning. If two bodies J/ and M^, Fig. 195, act upon each other 

with the forces P and F^, the directions 
of which, do not coincide with that of the 
common normal Jl JTto the two surfaces 
of contact, a decomposition of the forces 
always occurs ; only that force N' or iV], 
whose direction is that of the normal, is 
transmitted from one body to the other, 
the other component force S or S^, on 
the contrary, remains in the body and 
must be counteracted by some other force 
or obstacle, Avhen the bodies are to be 
held in equilibrium. But according to 
the principle announced, the two normal 
components A^and JVi must be exactly 
equal. If the direction of the force P 
forms an angle JV A P = a with the normal A X and an angle 
S A P — (3 with the direction of the other component >S', we 
have (see § 78) 




262 



GENERAL PRINCIPLES OF MECHANICS. 



[^ t39. 



i\r 



P sin. 3 



:,S = 



P sin, a 



sin. {a + 3y ^ ~ sin. (a + [3)' 
Designating in like manner JYi Ai P^ by a^ and Si A^ P^ by /3i, 
we have also 

Pi sin. /3i ^ ^ Pj sin. a^ 

sin. (ai + /3i) ^ 



^1 = 



sin. (tti + j3i) 
and, finally, since N = Ni 

P sin. (3 



and Si = 



Pi sin.' jSi 



Fig. 196. 



sin. {a + (3) si?i. {a^ + fii)' 

Example. — How are the forces decomposed, when a body M^, Fig. 196, 

held fast by an impediment I) E, is pressed 
upon by another body M^ movable about 
its axis (7, with a force P = 250 pounds ? 
The angles formed by the directions are 
the following : 

PAIi= a = d5' 
P A 8 = (5 = 48" 




P, A^ N, = 



65 



K» 



Pi A, 



S^ = f3^ = 50". 
The normal pressure between the two 
bodies is determined by the first formula 
and is 

^ sin. (a + /?) 
250 s^•?^. 48" ^^^^^ 

from the second we have the pressure on the axis or bearing G 

P sin. a 250 sin. 35' ,,, ,^ 

8 = -. — y = . — -^~- — = 144,47 pounds ; 

svi. (a + /3) sm. 83' ^ • 

and, finally, by combining the third and fourth formula we obtain the 

component which presses against the impediment D E 



8, = 



iV^i sin. a^ 187,18 sin. 65" 



sin. jS^ 



sin. 50" 



= 221,46 pounds. 



§ 139. In consequence of the equality of action and reaction, 
the equilibrium of a supported body is not changed, when, instead 
of the support, we substitute a force, which counteracts the pressure 
or tension transmitted to the support, and which is, therefore, 
equal in magnitude and opposite in direction to it. After having 
introduced this force, any body supported or partially retained 
may be considered as entirely free, and consequently its state of 
equilibrium can be treated in the same manner as that of a free 
body or of a rigid system of forces. 



% 140 ] EQUILIBRIUM OF BODIES R[GIDLY FASTENED. 263 

If, E.G., a body M, Fig. 197, is movable around its axis C, the 
force N is transmitted to a second body J[f„ the force S is counter- 
acted by the axis C and we can assume, that the body is entirely 
free and that besides F two other forces — i\^and —S act upon it. 
If the body i/i presses upon if with the force iVi and against the 
fixed plane B E with the force /S',, the equilibrium would not be 
disturbed, if instead of these impediments we should substitute two 
opposite forces — iVi and — 8^ and combine the same with the 
forc3S (e.g. with i^i), which act upon the body. In a state of 
equilibrium the resultant of the forces in the one as well as that 




-^r 



^ -S' 



in the other body must be null, and therefore the resultant of 

— N and — S must be counteracted by P and the resultant of 

— iVi and — 8x by Pj. 

Since the forces iV^and iV",, with which the two bodies act upon 
each other, are in equilibrium, the forces P, — S, Pj and — Si 
must be in equilibrium, when the combination of the two bodies 
{M, J/i) is in equilibrium. The forces iV, N^ are called the interior 
and the forces P, — ^S', Pi and — Si the exterior or extraneous 
forces of the combination of bodies or of the system of forces, and 
we can therefore assert that not only the interior forces are in eqtii- 
nhrium, hut that the exterior forces are so also, when, as is repre- 
sented in Fig. 198, we suppose the forces applied in any point 0. 

tv 

§ 140. Stability. — When a body supported upon a horizontal 

piano is acted on by no other force 'than that of gravity, it has no 
tendency to move forwards ; for its weight, acting vertically down- 
wards, is completely counteracted by this plane, but a rotation of 



264 



GENERAL PRmCIPLES OF MECHANICS. 



[§ 140. 




the body may be produced. If the body A D B F, Fig. 199, rests 
with the point D on the horizontal plane H R, it will remain at 

rest as long as its centre of gravity S is 

supported, i.e., as long as it lies in the 

vertical line (vertical line of gravity), 

passing through the point of support D. 

But if a body is supported in two points 

upon the horizontal surface of another 

body, the conditions of equilibrium 

require, that the vertical line of gravity 

shall pass through the line joining the 

two points of support. If, finally, a body 

rests upon three or more points on a horizontal plane, equilibrium 

exists, when the vertical line of gravity passes through the triangle 

or polygon formed by joining these points by straight lines. 

We must also distinguish for supported bodies, stable and un- 
stable equilibrium. The weight 6^ of a 
body A B, Fig. 200, draws the centre 
of gravity 8 of the same downwards; 
if there is no obstacle to tlie action 
of this force, it produces a rotation of 
the body, which continues until the 
centre of gravity has assumed its lowest 
position and the body has assumed a 
state of equilibrium. We can assert 
that the equilibrium is stable, when the 
^ centre of gravity occupies its lowest position (Fig. 201), that it is 
unstable, when it occupies its highest position (Fig. 202), and that 
Fig. 201. Fig. 202. Fig. 203. 

C B 



Fig. 200. 





finally the equilibrium is indifferent, when the centre of gravity re- 
mains at the same height, no matter what may be the position of 
the body (Fig. 203). 



§ 141.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



265 




Examples. — 1) The homogeneous body AD B F, Fig. 204, composed 
of a hemisphere and a cylinder, rests upon a horizontal plane H B. Re- 
quired the height S F = h of the cylindri- 
cal portion in order that this body shall be 
in equilibrium. Any radius of a sphere is 
perpendicular to the tangent plane corre- 
sponding to it, but the horizontal plane is 
such a plane, and consequently the radius 
SD must be perpendicular to it and contain 
the centre of gravity. The axis F 8 L 
passing through the centre of tlie sphere is 
also a line of gravity ; the centre 8^ as inter- 
section of the two lines of gravity, is therefore the centre ot gravity of the 
body. If we put the radius of the sphere and of the cylinder 8 A=: 
8B = 8 L = r, and the altitude of the cylinder 8 F = B F = h, we have 
for the volume of the hemisphere V^ = ^ n r^, and for the volume of the 
cylinder V^ = tt r^ h^ for the distance of the centre of gravity of the sphere 
/S'l, 8 8^ = f ^ and for that of the centre of gravity of the cylinder 
821 S 8^ = ^ h. In order that the centre of gravity of the whole body fall 
in 8 we must make the moment of the hemisphere f tt r^ . f r equal to the 
moment of the cylinder tz r^ h .^h, whence we have 

^2 = 1 r^ or /^ = r Vf = 0,7071 r. 
If the body is not homogeneous, but on the contrary the hemispherical 
portion has the specific gravity e^ and the cylindrical portion the specific 
gravity eg, then the moments of these portions are f rr r'^ • < 1 f ^ and 



. I" A, and consequently by equating them we have 



r. 



2) The pressure, which each of three legs ^, B, C, Fig. 205, of an arbi- 
F16 205 trarily loaded table has to bear, can be 

determined in the following manner. 
Let 8 be the centre of gravity of the 
loaded table, and 8 F, C B perpendicu- 
lars upon A B. Designating the weight 
of the entire table by O and the pres- 
sure in G by i?, we can treat A B as an. 
axis and put the moment of 72 = the mo- 
ment of G, I.E., B.G D= Q.8E, 
from which we obtain 
j._8E i^AB8 

and in like manner for the pressure in B^ we have 

t. AG8 




Q = 



P = 



A^ CB 
aB^G8 
A A BG 



G, and for that in A 



G, 



366 



GENERAL PRINCIPLES OF MECHANICS. [§ 141, 143. 



§ 141. Let us now investigate more fully the case of a body 
resting with one base upon a horizontal plane. Such a body pos- 
sesses stability or is in stable equilibrium, when its centre of gravity 
is supported, i.e. when the vertical line passing through its centre 
of gravity passes also through its base, since in this case the rota- 
tion, which the weight of the body tends to produce, is prevented 
by the resistance of the body. If the vertical line passes through 
the periphery of the base, the body is in unstable equilibrium; and 
if it passes outside of the base, the body is not in equilibrium, but 
will rotate around one of the sides of the periphery of its base and 
be overturned. The triangular prism ABC, Fig. 206, is conse- 
quently in stable equilibrium, since the vertical line S G passes 
through a point ^^of its base B C. The parallelopipedon A B CD, 
Pig. 207, is in unstable equilibrium, because t?ie vertical line S G 
passes through one of the edges D of the base C D. Finally, the 
cylinder A B C D, Fig. 208, is without stability; for S G does not 
pass through its base CD. 

Fig. 206. Fig. 207. 





Fig. 208. 
B 



Stability (Fr. stabilite, Ger. Stabilitat or Standfahigkeit) is the 

capacity of a body to maintain by 
its weight alone its position and 
to resist any cause of rotation. If 
we wish to select a measure for the 
stability of a body, it is necessary 
to distinguish the case of simply 
moving the body from that of 
actually overturning it. Let us 
first consider the former case 
alone. 

§ 142. Formulas for Stability. — A force P whose direction 
is not vertical tends not only to overturn, but also to push forward 
the body A B D, Fig. 209. Let us suppose that there is an 




^ 143.] EQUILIBRIUM OF BODIES RiaiDLY FASTENED. 



267 



Fig. 309. 




obstacle to its pushing or pulling the body forwards, and let us 
consider only the rotation around an edge C. If from this edge 
we let fall a perpendicular G E — a upon the direction of the 

force and another perpendicular 
C N = e upon the vertical line of 
gravity S G- of the body, we have 
then a bent lever E G N,to which 

the formula Pa— G e oy P — - G 

a 

is applicable. If, therefore, the ex- 
terior force P is slightly greater than 

Ge 

— , the body begins to turn around 

C and thus loses its stability. Its stability is therefore dependent 
upon the product {G e) of the weight of the body and the smallest 
distance of a side of the periphery of the base from the vertical line 
passing through the centre of gravity, and G e can therefore be 
considered as a measure of stability, and we will henceforth call it 
simply the stability. Hence we see that the stability increases 
equally with the weight G and with the distance e, and conse- 
quently we can conclude that under the same circumstances a wall, 
etc., whose weight is two or three tons, does not possess any more 
stability than one, whose weight is one ton and in which the dis- 
tance or arm of the lever e is two or three fold. 

§ 143. 1) The weight of a parallelopipedon.^1 B G D, Fig. 210, 
whose length is I, whose breadth \^ A B ~ C D = b and whose 
height is A D =. B G = h,is G = Vy = b lily, and its stability 

Gb 



St ^ G.D N= G ,IGD =^ 



\¥lil y, 

y denoting the heaviness of the material of the parallelopipedon. 
Fig. 210. Fig. 311. 

A B A B 





G - G 

2) The stabilities of a body B D E, Fig. 211, composed of two 



268 GENERAL PRINCIPLES OF MECHANICS. [§143. 

parallelopipedons, in reference to the two edges of the base C and 
F, are different from each other. If the heights are B C and E F 
= h and hi and the widths D and D F = b and bi, we have the 
weights G and Gi of the two portions = b h I y and bi hi I y; the 
arms in reference to C are C JY = -^ b and C = b + ^ bi, and 
those in reference to i^are bi •{• j^b and -^- bi, and the stabihty its, 
first, for a rotation around C 

St = iGb+ Gi{b + ^bi),= {^¥n + bbih -hibi'hi)ly, 
and, secondly, for a rotation about F 

Sti= G{bi+ \b) +iGibi = {Ui'hi + bbih + -lb'h)ly. 
The latter stability is Sti — St — {h — hi) bbily greater than 
the former. If we wish to increase the stability of a wall A by 
offsets D F, we must put them upon the side of the wall, towards 
which the force of rotation (wind, water, pressure of earth, etc.) 
acts. The stability of a wall A B C E, Fig. 212, which is battered 

on one side, is determined as follows. Let 

the length of the wall be I, the width on 

top A B =b, the height B G =h and the 

batter = n, i.e. when the height A K = 

1 foot the batter K L ^= n, or for a height 

h feet, = n h. The weight of the parallel- 

opipedon A G is G = b h I y, that of the 

triangular prism A D E ~ Gi — I n h . 

hly\ the arms for a rotation about E are 

E N= ED -\- ^b^nh -V Ib^Ti^EO 

^= % E D ^= \nh. Hence the stability is 

St^G{nh-\-\b)-Vl Ginh = {W + nhb + {n' ¥)hly. 

A parallelopipedical wall of the same volume isb + ^nh wide, 

and its stability is 

Sti = l{b + ^^nhyhly={^b'' + \nhb + ln'h'')hly', 
the stability is therefore St — Sti = (b -h j\ n h) , i n h"" I y 
smaller than that of a battering wall. 

The stability of a wall with a batter on the other side is 
St, = {¥ + nhb + It^h') .Ihl y, , 
and consequently smaller than St by an amount 
St -St, = {b + inh).inhUy, 
but greater by an amount St, — Sti = 2^4 ^^ ^^ I 7 than the sta- 
bility of a parallelopipedical wall of the same volume. 

Example. — What is the stability per running foot of a stone wall 10 
feet high, 1^ feet wide on top and with a batter of ^ of a foot on its back ? 
The density of this wall can be put (§ 61) = 3,4, consequently its heaviness 




§144.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 269 

is y = 62,4 . 2,4 = 149,76 pounds ; but we have ? = 1, ;^ = 10, 5 = 1,25 
and n = ^ = 0,2, and consequently the required stability is 
J3t=[^. (1,25)^ + 0,2 . 1,25 . 10 + ^ (0,2)^ . 10^] 10 . 1 . 149,76 

= (0,78125 + 2,5 + 1,3333) 1497,6 = 4,6146 . 1497,6 = 6911 foot-pounds. 

If the same quantity of materials is used, under the same circumstances 
the stability of a parallelopipedical wall would be 
St^ = [^ . (1,25)' + ^ . 0,2 . 1,25 . 10 + i (0,2)^ . 10^] . 149,76. 10 

= (0,78125 + 1,25 + 0,5) 1497,6 = 2,531 . 1497,6 = 3790 foot-pounds. 

The stability of the same wall with a batter on its front would be 
j8t^ = [1 (1,25)^ + 1. . 0,2 . 1,25 .10 + 1 (0.2)^ . 102] 149,76 . 10 

= (0,78125 + 1,25 + 0,666) 1497,6 = 2,6979 . 1497,6 = 4040 foot-pounds. 

Remabk. — We see from the above that we economize material by bat- 
tering the wall, by furnishing it with counterforts or offsets, by building 
it on plinths, etc. This subject will be treated more in detail in the second 
volume, where the pressure of earth, arches, bridges, etc., will be con- 
sidered. 

2> ■ :- ^'^" 

§ 144. Dynamical Stability. — We must distinguish from 
the measure of stajbility given in the last paragraph another meas- 
ure of the stability of a body, in which we bring into consideration 
the mechanical effect necessary to overturn the body. The work 
done is equal to the product of the force and the space ; the force 
in a heavy body is its weight, and the space is the vertical pro- 
jection of the space described by the centre of gravity, and, con- 
sequently, in the latter sense the product G s can be employed as 
the measure of the stability of a body, when s is the vertical height, 
which the centre of gravity of the body must rise, in order to bring 
the body from its state of stable into one of unstable equilibrium. 
Let C be the axis of rotation and S the centre of gravity of a 

body A B CD, Fig. 213, whose dy- 
FiG. 213. namical stability is to be deter- 

/'\ mined. If we cause the body to 

rotate, so that its centre of gravity 
S comes to S^, i.e. vertically above 
Cf the body is in unstable equili- 
brium ; for if it is caused to revolve 
a little more, it will tumble over. 

If we draw the horizontal line 
S JV, it will cut off the height 
JS/'Si= s, which the centre of gravity 
has ascended, by the aid of which we obtain the dynamical sta- 
bility G s. If now we have C S = C S, = r, C M = JSf S = e 
and the altitude C N = M S = a, we obtain 




270 



GENERAL PRINCIPLES OF MECHANICS. 



[§144. 



and the stability in the second sense is 
St = O (V^' 

Va 



+ e" — a, 



+ e' — a). 

The factor s = Va" -{- e' — a gives, for ^ = 0, 5 = e, for a — e^ 
e ( V2 — 1) = 0.414 e,ioxa = n e, s = { Vn^ + 1 — n) e, ap- 



proximatively =: {n -{■ 



2n 



n) e = TT-, thus for a = 10 e, s 
' 2n 



e 

20 



and for a— co ,s 



e 

00 



; this stability, therefore, becomes greater 



and greater as the centre of gravity becomes lower and lower, and 
it approaches more and more to zero as the centre of gravity is 
elevated more and more above the base. Sleds, wagons, ships etc. 
should therefore be loaded in such a manner, that the centre of 
gravity shall lie not only as low as possible, but also as near as 
possible above the centre of the base. 

If the body is a prism with a symmetrical trapezoidal section, 
such as is represented in Fig. 213, and if the dimensions are the 
following : length = I, height M = h, lower breadth C D = h^, 
upper breadth A B = l.^, we have 



a 



MS ^■ 
CM= e 

CS=r 



br + b, 3 

J b^, whence 






and the dynamical stability or the mechanical effect necessary to 
overturn this body is 

St = G ^' '^'' ■ 7^' + 2 Z>2 hy b, + 2 b, Ti 



^ It) + vxTX" • 3) 



-1 

•3J 




+ h '3/ &1 + b. 

What is the stability of, or what is the mechanical effect 

necessary to overturn, the granite obelisk A B C D^ 

Fig. 214, when its height is y^ = 30 feet, its upper length 

and breadth Z^ = 1|^ and J^ = 1 foot and its lower 

length and breadth Z3 = 4 feet and Jg = 3|^ feet ? The 

volume of this body is 

h 
F= (2 &i ?! + 2 §2 Zg + &i Z3 + &2 I,) g 

= (2 . f . 1 + 2 . 4 . f + 1 . 4 + I . f ) ^0 
= 40,25 . 5 = 201,25 cubic feet. 
If a cubic foot of granite weighs 7 = 8. 62,4 = 187.2 
pounds, we have for the total weight of the body 
a = 201,25 . 187,2 = 37674. 
The height of its centre of gravity above the base is 



§145.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 271 



a = 



&g ^2 + 3 ^ 1 ^1 + &2 ^1 + &i ^ 2 ^ 
2 ^3 ^2 + 2"6j Zj +^2^1 + &i ^2 ' ^ 

4.^ + 3.f .1 + 1.4 + 1.-51 



30 — 
"2"" — 



27,75 . 15 



10,342 feet. 



40,25 • ^ 40,25 

Supposing a rotation around the longer edge of the base, we have the 
horizontal distance of the centre of gravity from this edge, e — \})^ = 
1. . |. = |- feet, and therefore the distance of the centre of gravity from the 
axis is 



C /S' = r = Va^ + e^ = V(l,75)^ + (10,342)^ ^ Vil0,002 = 10,489 ; 
hence the height that centre of gravity must be lifted is 

8 = r - a = 10,489 - 10,342 = 0,147 feet, 
and the work to be done or tlie stability 

8t= Gs= 37674 . 0,147 = 5538 foot-pounds. 



■^.1 



Fig. 215. 



§ 145. Work Done in Moving a Heavy Body. — In order 
to find the mechanical effect, which is necessary to change the 
position of a heavy body by causing a rotation, we must pursue the 
same course as in calculating its dynamical stability. If we cause 
a heavy body A C, Fig. 215, to rotate about a horizontal axis to 
such an extent, that the inclination M C 8 = a of the line of 

gravity C S — r becomes M O 8i = a,, 
the centre of gravity S will describe the 
vertical space H S^ = M^ 8i — M S — s^ 
— r {sin. a, — sin. a), and therefore if we 
designate by G the weight of the body, 
the mechanical effect required is 

^1 = G Si =^ G r (sin a^ — sin. a). 

If the axis of rotation is not horizon- 
tal, but inclined at an angle 13 to the 
horizon, we have 

s^ = r cos. (3 {sin. Oj — si7i. a) and 

Ax = G S]= G r COS. (3 {sin. a^ — sin. a). (Compare § 133.) 
If in addition the body is moved in such a manner as not to 
change its position in relation to the direction of gravity, and if its 
centre of gravity and all its parts describe one and the same space, 
the vertical projection of which is — s^, then the moving of the 
body will require, in addition to the above mechanical effect, an 
amount of work A^—G 5^, and consequently the total work done 




will be 



A = Ax -\- Aci = G \r COS. (3 (sin. a^ — sin. a) + ^a-] 



The space described by the body in a horizontal direction does 



*v-,- 


A ,, 




; 


\ 


""^W^ 


/ ! 


t • 


'^^P 


1 1 

1 1 


r' D 


I 


( 1 


/\ O ^ 




/ s4- 


^\ 


■v» 


/ 1 ' 


N \ 


^ 


/ "r 


,.X---l- 

\5 


*<. 




272 GENERAL PRINCIPLES OF MECHANICS. [§ 14G 

not enter into the question, when we suppose the motion to he slow, 
in which case the work of inertia can be put equal to zero. 

If a body A G, Fig. 216, l3ring upon a horizontal plane B C is to 
be placed upright upon another plane Oc^ D^, we have (3 = 0% or 

COS. jS = 1 ; and if a and e 
Fig. 216. denote the horizontal and 

i..^...^z yertical co-ordinates of the 

centre of gravity of the body, 
when it is in an upright 
position, the radius C Si = 

r = Va^ + e^, and the height 
^i Si = a = r sin, a^. If a 
is the angle of inclination 
BOS formed by the side 
B (7of the body with the line 
of gravity C S, we have the 
original height of the centre 
of gravity above the surface 
on which the body rests 

K S =■ C S sin. B CS = r sin. a — Va^ + e" sin. a, 

and consequently the height, which the centre of gravity is raised, 
while the body is being placed upright is 

HS, = s, = E,Si'- EiH=a- Va' + e'sin.a.. 
If now ^2 is the vertical distance of the plane Cj B^ above the 
first plane B C, we have for the entire work done in placing the 
body upon C\ D^ 

A = G (a ~ Va" + e^ sin. a + Sa). 
This determination of the work necessary to move the body is 
perfectly correct only, when the centre of gravity is raised by a con- 
tinuous movement from S to S^. If, on the contrary, the body is 
first placed upright and then raised, the mechanical effect neces- 
sary is 

A^G{FO + s,)=a{U^-KS+s,) = a[Va^T~e^{l-sin.a)+s.;\', 



/_ 




for the work G . JV^ which the body performs, when the centre of 
gravity sinks from to >S'i,is lost. 

§ 146. Stability of a Body on an Inclined Plane. — ^A body 
A G, Fig. 217, resting upon an inclined plane (Fr. plan incline, 
Oer. schiefe Ebene), can assume two motions ; it can slide down 



§146.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



273 



Fig. 317. 



the inclined plane, or it can overturn by a revolution around one 
of the edges of its base. If the body is left to itself the weight G is 
decomposed into a force iV^at right angles to and a force P parallel to 
the base ; the first is counteracted entirely by the inclined plane, 
the latter, however, moves the body down the plane. If we put 
the angle of inclination of the plane to the horizon = a, we have 

also the angle G S J^ = a, and 
consequently the normal pressure 
^ ^ N = G COS. a and 

the sliding force 

P = G sin. a. 

If the vertical line of gravity 

>S' G passes through the base D, 

as is shown in Fig. 217, the sliding 

motion alone can take place ; but 

if the line of gravity, as in Fig. 218, passes without the base, the 

body will be overturned and is without stability. 

The stability of a body A C upon an inclined plane F H, Fig. 






11^ 



219, is different from that of a body upon a horizontal plane H R. 
li D M — e and M s = a are the rectangular co-ordinates of the 
centre of gravity S, we have for the arm of the stability 
DE=DO — MN:=e cos. a — a sin. a, 
while, on the contrary, it is = e, when the body stands upon a hori- 
zontal plane. Since e > e cos. a — a sin. a, the stability in refer- 
ence to the lower edge D is always smaller upon the inclined plane, 

and become null, when e cos. a — a sin. a, i.e. when tang, a = -. 

If, then, a body, whose stability h G e when standing upon a hori- 
zontal plane, is placed upon an inclined plane, whose angle of incli- 

nation corresponds to the expression tang, a = -, it loses its sta- 

18 



274: 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 147. 



bility. On the other hand, a body can acquire stability upon an 
inclined plane, although wanting it when placed upon a horizontal 
one. Eor a rotation about the upper edge (7 the arm is C Ei= C 0^ 
+ M N — Sx COS. a + a sin. a, while for the same position on a 
horizontal plane it is C 31 = e^. If, however, ei is negative, the 
body possesses no stability as long as it rests upon a horizontal 
plane ; but if placed upon an inclined plane, the angle of inclina- 

tion a of which is such that we have tang, a > ^, the body acquires 

a position of stable equilibrium. If, in addition to the force of 
gravity, another force F acts upon the body A B O D, Fig. 209, it 
retains its stability, if the direction of the resultant J\^ of the weight 
G of the body and of the force F passes through the base C D 
of the body. 

Example.— In the obelisk in the example of paragraph 144, e = |- and 
a = 10,342 feet ; consequently it will lose its stability, when placed upon 
an inclined plane, for whose angle of inclination we have 

7 7000 

'^^^- " = 47io;342 = 41368 = ^'^^^^^' 
and whose angle of inclination is therefore 

a = r 36'. 

§ 147. Theory of the Inclined Plane. — Since the inclined 

plane counteracts only the pressure 
perpendicular to it, the force P, ne- 
cessary to retain the body, which is 
prevented from turning over, on the 
inclined plane, is determined by the 
consideration, that the resultant iV, 
Fig. 220, of F and G must be per- 
pendicular to the inclined plane. Ac- 
cording to the theory of the parallel- 
ogram of forces, we have 

_ sm. F N ^ 

but the angle F N = angle (? iV^ = i^ ^ i^ = a, and the 
angle F N = F OK^KON^(i + 90°, when we denote 
the angle F E F — F K formed by the direction of the force 
with the inclined plane by i3 ; hence we have 

F sin. a F _ si7i. a 

'G ^ smTO^ + 90)' ^'^' 1} ~ co^' 




§ 148.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



275 



and the force, which holds the body on the inclined plane, is 

_ (t sin. a 
~ COS. fi 
For the normal pressure we have 

N^ _ sin. G JSr 

7} ~ sin. N G' 

or, since the angle 6^ iV^ = 90° — (a + /3) and iV (7 = P iV 

= 90 + ^, 

N _ sin. [900 - (g + /3)] _ cos, (a ,+ (i) 
'G'~ sm. (900+/3) ~ cos. (i ' 

and the normal pressure against the inclined plane is 

G cos. {a -^^ (i) 



N 



COS. (3 



Fig. 321. 
P 



If a + i3 is > 90° or jS > 90°— a, N becomes negative, and 
then, as is represented in Fig. 221, the inclined 
plane H F must be placed above the body 0, to 
which the force P is applied. If the force P is 
parallel to the inclined plane, (3 becomes = and 
COS. i3 = 1, and we have 

P = G sin. a and N ^=^ G cos. a. 
If the force P acts vertically a + /3 is = 90% 
and we have 

COS. (3 = sin. a, cos. (a + /3) = 0, 
P z=i G and JV = 0. In this case the inclined 
plane has no influence upon the body. 

Finally, if the force is horizontal, /3 becomes =-• — a and cos. (3 
= COS. a, and we have 
G sin. a 




P = 



COS. a 



= G tang, a and iV^ 



G COS. 
COS. a 



G 



COS. a 



Example. — In order to retain a body weighing 500 pounds upon a 
plane inchned to the horizon at an angle of 50°, a force is employed, whose 
direction forms an angle of 75° with the horizon : required the intensity of 
the force and the pressure of the body upon the inclined plane. The 
force is 

500 sin. 50" 500 »in. 50'^ ,^^ „ 

P — --— -— — = — = 422,6 pounds. 

COS. (7o" — 50") COS. 25" ' ^ 

and the pressure upon the plane is 

500 COS. 75" 

- 142,8 pounds. 



>M^\' 



N = 



COS. 25" 



§ 148. The Principle of Virtual Velocities. — If we com- 
bine the principle of the equality of action and reaction, explained 



376 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 148. 



Fig. 232. 






in § 138, with the principle of virtual velocities (§ 83 and § 98), we 
obtain the following law. If two bodies J/j and J/g hold each other 
in equilibrium, then, for a finite rectilinear or for an infinitely/ small 
curvilinear motion of the point A of pressure or contact, not onlt/ 
the sum of the mechanical effects of the forces of each separate 

hocly, hut also the sum of the me^ 
chanical effects of the exterior 
forces acting upon the tivo bodies 
{tahen together) is equal to zero. 
If Pi and Si are the forces in 
one body and Pa and /S'2 those 
in the other, when the point of 
contact is moved from A to B, 
the spaces described by these 
forces are A Di,A E^, A P2 and 
A E<i, and according to the law 
announced above we have 




P, , A D, + S, . A E, + P, A D, + 8, > A E, = 0, 
or without reference to the direction 

Pi AD\ + 8, . A~E, = P, . AP, + 8, . AE^. 
The correctness of this law can be demonstrated as follows. 
Since the normal forces iV"i and iVg are equal, their mechanical 

effects lij . A C and N'q ' A C must also be equal to each other, the 
only difference being, that one of the forces is positive and the 
other negative. But according to what we have already seen, the 
mechanical effect of the resultant JV^ . A ~C is equal to the sum 
of those Pi A1)i + 81 . A El of its components, and in like man- 
ner iVs -4 C = Pci. A P^ + 82 . A E^; consequently we have 

Pi . APi + 81 . aTEi = p, . aTd, + 8, . aTe,. 

This more general application of the principle of virtual 

velocities is of great importance in 
researches in statics, the determina- 
tion of formulas for equilibrium be- 
ing much simplified by it. If, e.g.,, 
we move a body A upon an inclined 
plane, FIT, Fig. 223, a distance A B, 
the space described by its weight G} 
is = ^ C=ABsin.AB = 

A B sin, F H R = A B sin a,, 
and, on the contrary, the space de- 



Fig. 223. 




149.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED. 



277 



scribed by the force P is = AD = AB cos. B A D = A B cos. (3, and 
finally that described by the normal force iV is = ; but the work 
done by JV is equal to the work done by G plus the work done by 
P, and we can therefore put 



]V.O=~G.AC+P.AD, 

consequently the force, which holds the body upon an inclined 
plane, is 

r, A C ^ G sin. a 



A D 



cos. (3 ' 



Fig. 224. 



a result, which agrees perfectly with that obtained in the foregoing 
paragraph. 

On the contrary, to find the 

normal force iV, we must move 
the inclined plane I£ F, Fig. 224, 
an arbitrary distance A B Ort 
right angles to the direction 
of the I'orce P, determine the 
space described by the exterior 
forces and then put the me- 
chanical effect of the weight G 
and of the force F equal to the 
mechanical effect of the pressure 
iV^upon the inclined plane. 
The space described by JV is 

A D = AB cos. BAD = AB cos. 13 ; 
that described by 6^ is 

A C=A B cos. B A C = A B cos. {a + (3), 

and that described by the force P is = 0, hence the mechanical 
effect is 

N . AD =^ G . ATC + P . 0, 




and 



-j^ _ G. AC _ cos. (a + j3) 
~ ~~AD~ ~ ^ * cos. (3 ' 



as we found in the foregoing paragraph. 

§ 149. Theory of the Wedge. — We can now deduce very 
simply the theory of the wedge. The wedge (Fr. coin, Ger. Keil) is a 
movable inclined plane formed by a three-sided prism F If G, 



278 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 149. 



Fig. 225. The force K P = P acts generally at right angles to 
the hack F G oi the wedge and balances another force or weight 



Fig. 225. 




A Q = Q, which presses against a side F H of the wedge. If the 
angle, which measures the sharpness of the wedge, is F II G = a 
and the angle formed by the direction K P or A I) of the force 
with the side G Ris GFK=BAD = 6, and, finally, if the 
angle L A H formed by the direction of the load Q with the sid*: 
F His = (3, the spaces described, when the wedge is moved froii s 
the position F H G to the position F^ H^ Gi, are found in the fol- 
lowing manner. The space described by the wedge is 

AB^FF.^HH,, 

that described by the force is 

A D =z A B COS. B A D ^ A B cos. 8, 
and that described by the rod ^ Z or by the load Q is 

. ^ __ A B sin. A B C _ A B sin. a _ A B sin. a 
sin. A C B sin. HAG sin. 13 

On the contrary, the space described by the reaction R of the 
base F G as well as that described by the reaction corresponding 
to the pressure against the guides of the rod is = 0^ 

Now putting the sum of the mechanical effects of the exterior 
forces P, Q, E and Bi = 0, we have 

P . AD - Q . A^ + E.O-h E,.0 =0, 
from which we obtain the equation of condition 

Q.AC Q . A B sin. a 



P = 



Q sin. a 
sin. (3 COS. 6' 



AD A B COS. 6 sin. [3 

If the direction K F of the force passes through the edge H of 



a 



the wedge and bisects the angle FH G, we have ^ = n, and therefore 



149.] EQUILIBRIUM OF BODIES RIGIDLY FASTENED 



279 



F = 



Q sin. a 



sin. 13 COS. 



a 
2 



2 Q sm. ^ 

sin. fi 



If the direction of the force is parallel to the base or side G H, 
we have 6=0, and consequently 

jy _ Q sin. a 
sin. 13 ' 

and if the direction of the load is also perpendicular to the side 
F H, we have (3 = 90% and consequently 

P ~ Q sin. a. 

Example. — The sharpness F H G = a of a wedge is 25", the direction 
of the force is parallel to the base, and therefore d is = 0, and the load acts 
at right angles to the side F H^ i.e., /? is = 90° : required the relations of 
the force and load to each other ; in this case we have 

p 

1 - Q sin. a or — = sin. 25" = 0,4226. ^ 

If the load is Q = ldO pounds, the force is 

P = 130 . 0,4226 = 54,938 pounds. 

In order to move the load or rod a foot, the wedge must describe the 
space 



AB = 



A G 



2,3662 feet. 



Fm. 226. 



sm. a 0,4226 

Remark 1. The relation between the force P and the load Q of the 
wedge F G 11, Fig. 226, can be determined by the application of the 

parallelogram of forces in the 
following manner. The load 

upon the rod A Q — Q is de- 
composed into a component 

A JV = iV perpendicular to the 
side F H and into a component 

A 8 = S perpendicular to the 
axis of the rod. While 8 is 
counteracted by the guides of 

the rod, A N = N is transmit- 
ted to the wedge and combines 

there as yl^ N^ with the force 

K P = A^P = P of the wedge to form a resultant AxR = i?, whose 
direction must be perpendicular to the base G II of the wedge, in which 
case it will be transmitted completely to the support of the wedge. The 
parallelogram of forces A^ P R N^ gives 




280 GENERAL PRINCIPLES OF MECHANICS. [§150. 

P _ sin. B A^ iVj _ sin. F H G _ sin. a 
N^ ~ sin. A^^ R N^~ sin. P A^ B ~ cos, 6* 

and from the parallelogram of forces A H Q 8 we have 

^_ sin. J^ Q A _ sin. Q A 8 _ 1 ^ 
Q ~ sin. A N Q~ sin. LAH~ sin. /3 ' 

but since N^ is = iV, we obtain by multiplying these proportions together, 

P N" _ P _ sin. a 
'N ' (i~ Q~ sin. /? COS. d' 

p _ Q sin. a 

~ sin. (3 COS. (J' 

as was found in the large text of this paragraph. 

Remark 2. The theory of the lever, inclined plane and wedge wiU 
be discussed at length in the fifth chapter, when the influence of friction 
will also be taken into consideration. 



CHAPTEE IV. 

EQUILIBRIUM IN FUNICULAR MACHINES. 

§ 150. Funicular Machines. — We have previously considered 
the solid bodies to be perfectly rigid or stiff bodies (Fr. corps 
rigides ; Ger. starre or steife Korper) ; i.e., as bodies, whose vol- 
ume and form are unchanged by the action of exterior forces upon 
them. Very often in the practical application of mechanics the 
supposition, that bodies are perfectly rigid, is not permissible, and 
it becomes necessary, therefore, to consider these bodies in two 
other states. These states are those of perfect flexibility and 
of perfect elasticity, and consequently we distinguish flexible 
bodies (Fr. corps flexible; Ger. biegsame Korper) and elastic 
bodies (Fr. corps elastiques ; Ger. elastische Korper). Flexible 
bodies counteract without change of form forces in one direction 
only and follow perfectly those acting in other directions ; elastic 
bodies, on the contrary, yield to a certain extent to every force 
acting upon them. 

A rigid body A B, Fig. 227, I, counteracts completely the force 



g 151.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



281 



P, a flexible body A B, Fig. 227, II, follows the direction of the 
force P, which acts upon it, in such a manner, that its axis assumes 




the direction of the force, and an elastic body A B, Fig. 227, III, 
resists the force P to a certain extent only, so that its axis under- 
goes a certain deflection. Cords, ropes, straps and in a certain 
sense chains are representatives of flexible bodies, although they do 
not possess perfect flexibility. These bodies will be the subject of 
the present chapter ; elastic bodies, or rather the elasticity of rigid 
bodies, will be treated of in the fourth section. 

We understand by a funicular machine (Fr. machine funicu- 
laire; Ger. Seilmaschine) a cord or a combination of cords (the 
word cord being employed in a general sense), which is stretched 
by forces, and we will occupy ourselves in this chapter with the 
theory of the equilibrium of this machine. The point of the 
funiculaire machine to which a force is applied, and where, conse- 
quently, the cord forms an angle or undergoes a change of direc- 
tion is called a knot (Fr. noeud ; Ger. Knoten). The same is either 
■fixed (Fr. fixe ; Ger. fest) or movable (Fr. coulant ; Ger. beweg- 
lich). Tension (Fr. tension ; Ger. Spannung) is the force propa- 
gated in the direction of its axis by a stretched cord. The ten- 
sions at the ends of a straight cord or piece of cord are equal and 
opposite (§ 86). A straight cord cannot propagate any other force 
but the tension acting in the direction of its axis ; for if it did, it 
would bend and would no longer be straight. 



§ 151, Equilibrium in a Knot. — Equilibrium exists in a 
funicular machine, when each of its knots is in equilibrium. Con- 
sequently we must begin with the study of the conditions of equi- 
librium in a single knot. 

Equihbrium exists in a knot K formed by a piece of cord 



283 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 151 



Fig. 238. 



A K B, Fig. 228, when the resultant K S = S oi the two tensions 

of the cord K Si = Si and KlS^ == S<i is equal and opposite to the 

force P applied at the knot; for the 
tensions of the cord Si and S^2 pro- 
duce the same effect in the knot 
K as two forces equal to them and 
acting in the same direction as 
they do, and the three forces are in 
equilibrium, when one of them is 
equal and opposite to the resultant 
of the other two (§87). In like 
manner the resultant E of the 
force P and of one of the tensions 
Si is equal and opposite to the 
second tension S.2, etc. We can 
profit by this equality to determine two conditions, e.g., the ten- 
sion and direction of one of the ropes. If, e.g., the force P, the 
tension Si and the angle formed by them 

A K P = 180"' - A XS = 180° - a 
are given, we have for the other tension 

s, = V p' + Si' - 2 p s^os:"a 

and for its direction or for the angle B K S = 13 formed by it 

withXS 

Si sin. a 




sin. 13 =: 



S<i 



Example. — If the rope A KB, Fig. 228, is fastened at its end B and 
stretched at its end JL by a weight G^ = 135 pounds and at its centre 
^by a force P = 109 pounds, whose direction is upwards at an angle of 
25° to the horizon, what will be the direction of the tension in the 
piece of cord K B"^. 

The intensity of the required tension is 



So = -J 109^ + 135^ - 2 . 109 . 135 cos. (90" — 25") 



= V 11881 + 18225 — 29430 . cos. 65° = V 17668,3 = 132,92 pounds. 
For the anojle fS we have 



ain. (3 = 



8^ sin. a _ 135 . sin. 65" 
^"^3 ^ 132^92 



, log »in. I3 = 0,96401 — 1, 



whence /? = 67° 0', and the inclination of the piece of cord to the horizon is 

(S° — 25° = 67° 0' — 25° 0' = 42° 0'. 



g 152.J 



EQUILIBRIUM IN FUNICULAR MACHINES. 



283 



§ 152. If a cord A K B, Fig. 229, forms a fixed knot A^in con- 
sefj^uence of one portion of the cord B K lying upon a firm sup- 
port My while the other portion of 

the cord is stretched by a force K 8 
= S, whose direction forms a certain 
angle S K Si = a with the direction 
of the first portion of the cord, we 
have the tension in the portion K B 
of the cord 




X Si = Si = S COS. a, 



while the second component K N — N =. S sin. a is counteracted 
by tlie support M. We have also 

Si = S Vl - {sin. ay, 
and therefore, when the angle of divergence is small, 

Si^[l-l {sin. a)^) .^ = (l - f) S, 

or inversely 

S=-^. = {l.-^)Si. 



1-- 

2 

If a cord is laid upon a prismatical body, and its directions thus 
changed successively an amount measured by the angles a^, a^, a^, 

the foregoing decomposition 
Fig. 330. Qf j^}^^ force is repeated, so 

that in the knot ^the ten- 
sion S is changed into Si = 
S COS. «!, and in the knot Kt 
the tension Si into 

S2—S1 COS. a^=S COS. «! COS. ttg, 

and in the knot ^3 the ten- 
sion S2 into 
S3 = S<i COS. a^ = S COS. «! COS. a^ cos. a^. 
If the angles «!, a^, a^ are equal to each other and = a, we have 

Ss = S {cos. a)' 
S„ = S {cos. a)". 
If the prism M becomes a cylinder, a is infinitely small and n 
infinitely great, and consequently 

or if we denote the total angle of divergence n ahy (3, we have 




384 GENERAL PRINCIPLES OF MECHANICS. [§153. 

^„ = (l - ''-^) S, I.E. 

Sn = S, because a and consequently -^ is infinitely small compared 

with 1. 

If, therefore, a cord is laid upon a smooth body so as to coyer a 
portion of the periphery of its cross section, its tension is not 
changed thereby ; and when a state of equilibrium exists the ten- 
sion at both ends of the cord are equal to each other. 

§ 153. If the knot K is movable, if, E.G., the force P is applied 
by means of a ring to the cord A K B, Fig. 231, which is passed 
through it, the resultant S of the tensions 8^ and ^2 of the cord is 
equal and opposite to the force P applied to the ring ; besides the 
tensions of the cord are equal to each other. This equality is a 
consequence of § 152, but it can also be proved in the following 
manner. If we pull the rope a certain distance through the ring, 
one of the tensions Sx describes the space s, the other tension Sc^ the 
space — s, and the force P the space 0. If, therefore, we assume 
perfect flexibility, the work done is 

P .0 = Si.s — S^.Sf I.E. SiS = Si sor Si = S^. 

The equality of the angles A K 8 and B K 8, formed by the 
direction of the resultant aS' with the directions of the rope, is also a 
consequence of this equality of the tensions. Putting this angle 
= a the resolution of the rhomb K 81 8 8^ gives 
8 = P = 2 81 cos, a, and inversely 

p 

8x = 8,= 



2 cos, a 



Fig. 231. 



Fig. 232. 





If A and B, Fig. 232, are fixed points of a cord ^ ^ ^ of a 



§153. 



EQUILIBRIUM IN FUNICULAR MACHINES. 



285 



given length (2 a) with a movable knot K, we can find the posi- 
tions of this knot by constructing an ellipse, whose foci are at A and 
B and whose major axis is equal to the length of the rope 2 a, 
and by drawing a tangent to this curve perpendicular to the given 
direction of the force. The point of tangency thus found is the 
position of the knot ; for the normal K S to the ellipse forms equal 
angles with the radii vectores X A and K B, exactly as the result- 
ant S does with the tensions S^ and S<i of the cord. 

If we draw A D parallel to the direction of the given force, 
make B D equal to the given length of the cord, divide A D in 
two equal parts at M and erect the perpendicular M K, we obtain 
the position K of the knot without constructing an ellipse ; for the 
angle A K M = angle D K M and A K = D K, and consequently 
the angle A K 8= angle BK8QXidiAK+EB = DK + 
KB = DB. 

Example. — Between the points A and B^ Fig. 233, a cord 9 feet long is 
■tretched by a weight G^ = 170 pounds, hung upon it by means of a ring. 

The horizontal distance of the two points 
from each other is ^ (7=6^ feet and the 
vertical distance of the same C B = 2 feet : 
required the position of the knot as well as 
the tensions and directions of the two por- 
tions of the cord. From the length A B = 
9 feet as hypotbenuse and the horizontal 
distance A C = 6^ feet, we obtain the ver- 
tical line 
CD 



Fig. 233. 




r 












^ 


B 


^v 


/ 


/S 


M 


^\ 


V 


{ 


1 


\ 

3 


) 



V_92_- 6,52 = V 81 - 42,25 
= V 38,75 = 6,225 feet, 
and from this the base of the isosceles tri- 



angle BBK 



BB = GD-CB = 6,225 - 2 = 4,225 feet. 
On account of the similarity of the triangles B iTJif and B AG^Vft have 



BK=BK=: 



BM 



B A = 



4,225 . 9 



3,054 feet, 



B C 2 , 6,225 

whence 

AK= 9 — 3,054 = 5,946 feet. 

Hence for the angle a formed by the two portions of the cord with the ver- 
tical line we have 



COS. a = 



B M 2,1125 



B K 3.054 
and finally the tension in the cord is 

8. = «,= « 



= 0,6917, whence a = 46° 14', 



2 COS. « 2 . 0,6917 



= 122,9 poundB. 



286 



GENERAL PRINCIPLES OP MECHANICS. 



[§154, 



Fig. 234. 




§ 154. Equilibrium of a Funicular Polygon. — The con- 
ditions of equilibrium of a funicular polygon, i.e. of a stretched 

cord acted upon in different 
points by forces, are the same 
as those of the equilibrium 
of forces applied at the same 
point. Let A KB, Fig. 234, 1, 
be a cord stretched by the 

forcesP„P„P3, P4, A; Pi 
and Pg being applied in ^, 
P3 in K and P4 and P5 in B. 
Let us denote the tension of 
the portion of the cord A K 
by Sx and that of the portion 
B Khj S^y then we have Si 
as the resultant of the two 
forces Pi and Pg applied in A. 
Transferring the point of ap- 
plication of this tension from A to K, we have S^ as resultant of 
Si and Pj or of Pi, F^ and P3. Transferring the point of applica- 
tion of the force S^ from K to B, we have So as the resultant of P4 
and P5 ; now, since S^ is the resultant of Pi, P2, and P3, this system 
of forces is in equilibrium ; we can therefore assert, that if certain 
forces Pi, P2, P3, etc., of a funicular polygon are in equilibrium, 

they will also hold 
Fig. 335. each other in equi- 

librium, when they 
are applied without 
change of direction 
or intensity to a sin- 
gle point, E.G. to C 
(II). If the rope 
AKiK,...B, Fig. 
235, is stretched in 
the knots Ki, K^, 
etc., by the weights 
Gi, G^y etc., and if 
its extremities are 
held fast by the ver- 
tical forces Vi and 
F„ and by the hori- 




§ 155.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



287 



zontal forces ^i and H„<, the sum of the vertical forces. is 

V,+ V,-{G, + G,-\- G, + ...), 
and the sum of the horizontal forces is Hi — H^, The conditions 
of equilibrium require both these sums to be = 0, and therefore 
we have 

1) Fi + F„ = 6^1 + 6^2 + 6^3 + . . . and 

2) H, - ^„, I.E. 

the stim of the vertical forces or tensions at the extremities of the 
ropes of a funicular polygon stretched dy iveights is equal to the sum 
of weights hung upon it, and the horizontal tension at one extremity 
is equal and opposite to that at the other. 

If we prolong the directions of the tensions S^ and Sn at the 
extremities A and B, until they cut each other in C, and if we 
transfer the point of application of these tensions to this point, we 
obtain a single force P=Vi-\- Vn', for the horizontal forces H^ and H^ 
balance each other. Since this force balances the sum G^ + Gc^ -\- 
Gs + ... of the weights attached to it, the point of application or 
centre of gravity of these weights must be in the direction of this 
force, I.E. in the vertical line passing through C. 

155. From the tension Si of the first portion A K^ of the 

rope and from the 
angle of inclination 
Si A Hi = fliwe ob- 
tain the vertical ten- 
sion Vi = Si sin. «i 
and the horizontal 
tension Hi = Si cos. a^. 
If we transfer the 
Hi, point of application 
of these forces from 
^ to ^i, we have, in 
addition to them, 
the weight 6^,, v/hich 
acts vertically down- 
wards, and the verti- 
cal tension in the 
following portion 
Ki K<, of the rope is F2 = F — 6^i = Si sin. a, — 6^„ while the 
horizontal tension H^ — Hi = H remains unchanged. The two 
latter forces, when combined, give the axial tension of the second 
portion of the rope 



Fm. 235. 




288 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 155. 



and its inclination a^ is determined by the formula 

__ ^2 _ ^1 sin. tti — Gx 



tang. Oa 



tang, a^ = tang, d^ — 



Si COS. «! 



I.E. 



If' 



Transferring the point of application of V^ and IT, from K^ to 
^2, we have, by the addition of the weight G„ a new vertical force 

V^= V,~ G,= Vx - (Gx + G,) = Sx sin. ax - (Gx + G,), 
which is that of the third portion of the rope, while the horizontal 
force H^—H remains unchanged. The total tension in this third 
portion of the cord is 

and its angle of inclination a^ is determined by the formula 

, F3 Si sin. ax — (Gx -\- G^) 

tang, a^ = ^ = ^— ' % i.e. 

±t Si COS. ax 



tang, a^ = tang, ax — 



Gx + G, 
H 



For the angle of inclination of the fourth portion of the cord 
we have 

, , Gx + Gi + Gz 
tang, a^ = tang, ax 



H 



etc. 



If 



Gx -\' Gi + G3 



becomes > tang, ax or Gx + G^ + G3 y Vi. 



then tang, a^ and consequently a^ becomes negative, and the cor- 
responding side ^3 JTi of the polygon is no longer directed down- 
ward, but upward. The conditions are the same for any point, for 
which 6^1 + G^2 + ^3 + . . • is > Fi. 

The tensions Sx, Sci, /S3, etc., as well as the angles of mclination 
Ox, a<i, Og, etc., of the different portions of the rope can easily be 
represented geometrically. If we make the horizontal line C A — 

C B, Fig. 236, = the horizontal tension 
// and the vertical line C Kx— the vertical 
tension Vx at the point of suspension^, 
the hypothenuse A Kx will give the total 
tension Sx of the first portion of the rope, 
and the angle C A Kx the inclination of 
the same to the horizon. If, now, we 
lay off upon C Kx the weights Gx, 6^2, Gz, 
etc., as the divisions Kx K^, K^ K^, etc., 
and draw the transverse lines A K^,A Kz, 




^156.] EQUILIBRIUM IN FUNICULAR MACHINES. 289 

the latter will indicate the tensions of the different succeeding 
portions of the cord, and the angles G A K^j C A K^, etc., the 
angles of inclination ag, a^, etc., of these portions. 

§ 156. From the investigations in the foregoing paragraph we 
can deduce the following law for the equilibrium of a cord stretched 
by weights : 

1) The horizontal tension is i7i all parts of the cord one and the 

same, viz.: 

11= Si COS. tti = S,, COS. a„. 

2) The vertical tension in any portion is equal to the vertical 
tension of the cord at the end above if minus the sum of the weights 
suspended above it, or 

V,„ =: Fi — (6^1 + 6^2 + • • • G,n-.\). 

This law can be expressed more generally thus : The vertical 
tension in any point is equal to the tension in any other lower or 
higher point plus or minus the sum of the weights suspended be- 
tween them. 

If we know besides the weights the angle a^ and the horizontal 

tension H, we obtain the vertical tension at the extremity A by 

means of the formula 

Vi = H , tang, a^, 

and that at the extremity B is 

F„= (G^, 4- 6^2 + ... + 6^„) - V,. 
If, on the contrary, the two angles of inclination a, and a„ at 
the two points of suspension A and B are known, the horizontal 
and vertical tensions are determined in the following manner; 
we have 

K _ tang, a^ 

Vi tang, a* 

and therefore K = —, ^— -^. 

tang, a. 

But since Fj + Vn = Gi + G^ + . . . I.E., 

Itang, a, + tang. a,\ j^ ^ , ^f 

\ 1^^, ; V,= G^ + G...„ 

we have 

(Gi -«- G<i -{- . . .) tang. «! /^ ^ v sin. a, cos. a„ 

tang, a^ -f- tang. a„ ^ sm. (a, + a„) 

and 

-^ _ (Gx+G^ 4- . . . ) tang. «« _ /^ , /> , \ sin. a„ cos, a , 
tang, a^ + tang. a„ stn. (a, + a,j 

and consequently 

19 



290 



GENERAL PRINCIPLES OF MECHANICS. 



[§15a 



H = Vi cotg, «! = Vn cotg. a„ = (^i + 6^2 + • • . ) . ' / ^. 

' sm. («! + a„) 

If the two ends of the cord have the same inclination, we have 

C 4- C -^- -\- P 
Fi = Vn = — — ^ ^ ; then one end A carries as much 

as the other end B, 

These formulas are applicable to any pair of points or knots of 
the funicular polygon, when we substitute instead of G^ + G-2 + - • • 
the sum of the weights, etc., suspended to the cord between the 
two points. The vertical tensions of a cord, on which a weight G,n 
is hung and the angles of inclination of which are a^ and a^ + 1, are 
sin. a^ COS. a^ + 1 G„, 



V = G 



sin. (a„ 



a 



m -)- 



1) 1 + cotg. a,, tang. a^_^^ 



and 



Y — r - ^^' ^"^ + ^ ^^^* ^" 

' m 4- 1 ^ni ' 



Gr. 



sin. (a„, + a,„ + i) 1 + tang, a^ cotg. a„ + i* 
These laws are applicable to any funicular polygon stretched by 
parallel forces, when we substitute instead of the vertical the direc^ 
tion of the forces. 

Example. — The funicular polygon A K^ K^ K^ B, Fig. 337, is stretched 
by three weights G^ = 20, G^ = 30 and G^ = 16 pounds as well as by 

the horizontal force H^ = 25 
Fig. 237. pounds ; required the axial ten- 

sions, supposing the extremities 
A and B to have the same angle 
of inclination. The vertical ten- 
sions at the ends are equal and are 
G^ + G^ + G^ 




_ 20 + 30 + 16 

~ 2 



2 

33 pounds. 



The vertical tension of the 
second portion of the cord is 
Fg = V^-G^ = dd - 20 = 13 
pounds ; that of the third is, 
Vs = V^ - G^ (or G^ + G^ - FJ = 33 - 16 = 17 pounds. 
The angles of inclination a^ and a^ of these extremities are determined 
bv the formulas 

33 

tang, a^ = tang, a^ 



V 

H "= 25 ==^'^^' 



tnose of the second and third portions by the formulas 



tang, a^ ^ tang, a^ 
tang, a.^ = tang, a^ 






20 
1,32 — ^ = 0,52 and 



s = 1'^^ - i ^ «■"« ■' 



§ 157.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



291 



whence we have 

a^ = a^= 52° 51', a^ = 27° 28', a^ = 34° 18'. 
Finally the axial tensions are 

^^ _ ^^ ^ VFi' + J?' = V 33^ + 25^ = Vl714 = 41,40 pounds. 



= VK^ + H' 



V 13'^ + 25'' = V 794 = 18,18 pounds and 
: 30,23 pounds. 



^^3 = VFg^ + H' = Vl7^ + 25^ 

§ 157. The Parabola as Catenary. — Let us suppose, that 
the cord A C B, Fig. 238, is stretched by the weights G^,, G^^, G:^, 

etc., hung at equal horizontal 
distances from each other. Let 
us denote th? horizontal dis- 
tance A M between the point of 
suspension A and the lowest 
point C hj J) and the vertical 
distance C M hj a; let us also 
put the similarly placed co-ordi- 
nates of a point of the funicu- 
lar polygon JSf = y and C N 
= X. If the vertical tension in 

y 
^ is = V, that in is consequently = t • V, and therefore we 

have for the angle of inclination to the horizon N T = R Q 
= of the portion of the cord Q 

tang. "P = ^ - -g> 

in which H designates the horizontal tension. 











Fig. 


238. 




V M 


B 


\ 


K 





R 






N 


/ 


G 


\ 


Q 






/ 




^ 


-. 








C:\ 






C 




"'c^ 


a 




1 


1 





OR 



From this we obtain Q R ^^ OR. tang. 

y V 

J . -==, which is the difference of height of two neighboring corners 

-12 

of the funicular polygon. If we put y successively = R, 2 R, 
Z R, etc., the latter formula gives the difference of height of 
the first, second, third, etc., corners, counting from the lowest 
point upwards ; if now we add all these values, whose number we 
•can suppose to be = m, we obtain the height C N oi the point 
above the lowest point C. Here we have 



X = C N = 



V OR.-. 
H ' 



{OR-\-20R + 30R-\-...i- m . R) 



^ R\, , ^ ^ . V 

jj . — T— (1 + 2 + 3 + ... + 77?)=-^ 



m (m + 1) R^ 



1.2 b 

in accordance with the rule for summing an arithmetical series. 



292 



GENERAL PRINCIPLES OF MECHANICS. 



[§15Z 



Finally, putting H = —,wg obtain 



X 



m 
_ V m {m -\- 1) y^ 

~ H ' ¥^' ' P 

or substituting for the value of the tangent of the angle of inclina- 

V 
tion a of the end A of the rope tang, a = -^ 

Jti 

_ m {m + 1) y^ tang, a 
2 m' h 

If the number of the weights is very great, we can put m -h 1 
= m, and consequently 



X = 



X = 



a 



V 



r 



tang, a. 



r 

H ' 2h 21 
For X = a, y ^= h, and consequently we have 

V i _i tang, a 
H ~ ~ 
X _ y"^ 

~a~'¥' 



2 



2 



or more simply 



which is the equation of a parabola. 

If, therefore, an imponderable string is stretched by an infinite 
number of equal weights applied at equal horizontal distances from 
each other, the funicular polygon becomes a parabola. 

For the angle of inclination we have 



tang, (p = 



y 2 a 

2a 
tang, a — -^. 

The subtangent for the point is 



o <^ o ^ 



2x 

y 



and 



2x 



y^— = 2x = 2GK 

y 



N T= Ntang.(t> 
If the chains and rods of a chain bridge A B D F, Fig. 239, were 




without weight or very light in proportion to that of the loaded 
bridge D E F, the latter weights alone would have to be considered, 
and the chain A C B would form a parabola. 



§ 158.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



293 



Example. — The entire load of the chain bridge in Fig. 239 is G — 2 V 
= 320000 pounds, the sjDan is AB = 2d = 150 feet, the height of the arc 
C M = a = 15 feet ; required the tension and other conditions of the 
chain. The inclination of the chain to the horizon is determined by the 

formula 

2 a SO 2 

tang. a = -^= — = - = 0,4, whence a = 21° 48'. 

The vertical tension in each point of suspension is 
F= 1^ weight = 160000 pounds, 
the horizontal tension is 

B:= Vcotg. a = 160000 . ^ = 400000 pounds, 
and the total tension at one end is 



8 = -^ V + B'' = F V 1 + cotg.'' a = 160000 . fl + (^\ 

/~29 

= 160000 r ^ = 80000 V29 = 430813 pounds. 

§ 158. The Catenary. — If a perfectly flexible and inextensible 
cord, or a chain composed of short links, is stretched by its own 
weight, the axis of the same will form a curved line, which has re- 
ceived the name of the catenary curve (Fr. chainette, Gr. Ketten- 
linie). The strings, ropes, ribbons, chains, etc., which we meet 
with in practice, are imperfectly elastic and extensible, and conse- 
quently form curves, which only approach the catenary, but which 
can generally be treated as such. From what precedes we know, 
that the horizontal tension in the catenary is equal at all points, 
while, on the contrary, the vertical tension in one point is equal to 
the vertical tension in the point of attachment above it minus the 
weight of the portion of the chain between this point and the point 
of suspension. Since the vertical tension at the vertex, where the 
catenary is horizontal, is = 0, or since the vertical tension at the 

point of suspension is equal to 
the weight of the chain from 
the point of attachment to the 
vertex, the vertical tension in 
any point is equal to the weight 
of the portion of the chain or 
cord below it. 

If equal portions of the chain 
are equally heavy, the curve 
produced is the common cate- 
nary, which is the only one we 



Fig. 240. 




294 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 159. 



will discuss here. If a portion of the chain or cord one foot long 
weighs y, and if the arc corresponding to the co-ordinates C M — a 
and MA = b, Fig. 240, is ^ C ^ 1,wq haye for the weight of 
the portion A O of the chain G = I y. 

If, on the contrary, the length of the arc corresponding to the 
co-ordinates G N =^ x and iV^ = y is — 5, we have for the weight 
of this arc V — s y. Putting, finally, the length of a similar piece 
of chain, whose weight is equal to the horizontal tension H, = c, 
we have H — c y, and we have for the angles of inclination a and 
(j) in the points A and 

tang, a = tang, S A H = -^f — — = - and 

tang. = tang. N G T = — = — =- 

§ 159. If we make the horizontal line G H, Fig. 241, equal to 
the length c of the portion of the chain measuring the horizontal 
tension and G G equal to the length I of arc of the chain on one 
side, in accordance with § 155, the hypothenuse G H gives the 
intensity and direction of the tension of the cord at the point of 

suspension A ; for 

G G I 
tana. G H G = ^ ^r = - and 
^ G H c 

'^ GW + GH' = VJ 

V G' + Jff' = |/FT~? 



GR 

S 



+ c\ or 



Fig. 341. 



.y = G H,y, 
If we divide G G into equal parts and draw from H to the 

points of division 1, 2, 3, etc., straight 
lines, the latter give the intensity and di- 
rection of the tensions obtained by dividing 
the length of the arc of the chain A G into 
as many equal parts. For example, the line 
UK giYQS the magnitude and direction of 
the tension or tangent at the point of di- 
vision (P) of the arc A P G, since at this 

point the vertical tension = G K. y, while 
the horizontal tension is constant and = 
c . y, and therefore for this point we have 

GK'.y GK 

tana, o = = ^ ^r 

^ ' cy G 11 

as is really shown by the figure. 

This peculiarity of the catenary can be 

made use of to construct mechanically, approximatively correctly. 




i- 160.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



295 



this curve. After having divided the given length of the catenary 
to be constructed in very many equal parts and laid off the hne 
C H = c, which measures the horizontal tension, we draw the 
transverse lines HI, iJ 2, H ^, etc., and lay off on C H 2^ division 

C i of the arc of the curve as C a, pass through the point of division 

{a) thus obtained a parallel to the transverse line H 1 and cut off 

again from it a part a b = C 1. In like manner we draw through 

the point (i) thus obtained a parallel to the transverse line H2 and 

cut off from it b c = (71 equal to a division of the arc. Wc now 

draw through the new point (c) a parallel to ^ 3 and make c d 
equal to a division of the arc and continue in this way, until we 
have obtained the polygon C a h cdef. We now construct another 

polygon Oa(3y6e(j) hy drawing C a parallel to H 1, a 13 to 112, 

(iytoHd, etc., and by making C a -— a (3 = {3 y, etc., = 01 = 

12 = 23, etc. If, finally, we pass through the centre of the lines 
«a, Z»ft cy.../0a curve, we obtain approximatively the catenary 
required. 

For practical purposes we can often obtain accurately enough 
a catenary corresponding to given conditions, e.g. to a given width 
and height of the arc or to a given width and length of arc, etc., 
by hanging a chain with small links against a vertical wall. 

§ 160. Approximate Equation of the Catenary. — In 

many cases, and particularly in its application to architecture and 
machinery, the horizontal tension of the catenary is very great 
compared to its vertical one, and therefore the height of the arc is 
small, compared with its width. Under this assumption, an equa- 
tion for this curve can be found in the following manner : 

Let s denote the length, x the abscissa C JV and y the ordinate 

iV^ of a very low arc C 0, 
Fig. 242. We can, according 
to the remark upon page 298, 
put approximatively 

^=[^ + 1(^)1^' 

and therefore the vertical tension in a point (9 of a low arc of a 
catenary is 

2 




^=b-u;n^r 



.^96 GENERAL PRINCIPLES OF MECHANICS. [§160. 



and the tangent of the tangential angle 2" iV" = is 

If we divide the ordinate y into m equal parts, we find the 
portion R Q = W U of the abscissa X corresponding to such a 
division Ehy putting 

RQ=0~R tang. ^===^^-f[l+|(-)'} 
Since x is very small compared to y, we have approximatively 

R Q = R . -. Substituting now OR — — and successively for 2/ 

(y tit 

the values — , — , — , etc., we obtain one after the other the differ- 
m m m 

ent portions of x, the sum of which is 

,= J^(l +2 + 3 + ... + m) = -^ ™-(^(§157) = f ; 
cni"^ 'cm" 2 ^^ ■^ 2 c 

the latter equation is that of the parabola. 

If we proceed more accurately and substitute in the formula 



Q-n=o-n-.i[i.im 



instead of x, the value ~ just found, we obtain 

Putting y again successively equal to -^, —^, -^, etc., and 

lit tit tit 

instead of R, — , we obtain successively the different portions of 
X, and consequently their sum 

x= y^\y- (1 + 2 + 3+.. .+w)+^^ . i^)\v + 2'-\-^'-^„. + m')\ 
c mLm ^ b c \m/ J 

When the number of members is very great, the sum of the cardi- 
nal numbers 1 + 2 + 3. .. + m is = y ^^^ *^^ ^^^ ^^ ^^^^^ cubes 
is = ^ (see " Ingetiieur," page 88). Hence we have 

'■'''-2 c 2^c'~ 2cr^ 12-\cl V 
the equation of very powerfully stretched catenary. 



§160.] EQUILIBRIUM IN FUNICULAR MACHINES. 297 

By inversion we obtain 

y» = 2 c a; — -^ = 2 c a; — T?r~^ = 2 ex — ^, whence 

x4i C x/i C o 



2) y = y 2 c a; — 5-? or approximatiyely, 



2^= i^Tc^ 1 - 



X 



12cl 
The measure of the horizontal tension is given by the formula 

_ y^ y* _ y^ y* 4 x^ 

^ ~ 2^ "^ 2x.l2c' ~ 2x'^ 2ix ' ~y^' ^'^' 

ox . _ ^' ^ 
^^ ' - 2^ + 6- 

The tangential angle is determined by the formula 

=T['*i(in['-Mir]'- 

The formula for the rectification of the curve is 

.).=,[.. i(i)>,[-i(f)']- 

Example — 1) The length of the catenary for a width of arc 2 & = 16 
feet and for a height of arc a = 2| feet is 

.. = ..[.. |(|J] = ie.[..l(fy] 

= 16 + 16 . 0,065 = 17,04 feet; 
and the length of the portion of the chain, which measures the horizontal 
tension, is 

" = ^ + ^ = T + ^ = i^'» + «'*" = ^^'^^"«^' • 

the tangent of the angle of inclination at the point of suspension is 

, 2ar, l/r/Vn 5 r 1/5 Vl 5.1,03255 
tang.a^-^\l + ^^-) J = _.| i +^y J = ^- =0,6453..., 

whence the angle itself is a = 32" 50'. 

2) If a chain is 10 feet long and the width of span is ^ feet, the height 
of arc is 

a/~Ki '^'^ /"3 (10 - 9^) 9i ./3 19 /57 
= ViTSlT = 1,335 feet, 



298 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 160. 



and ttie measure of the horizontal tension is 



a 



4.752 



^■~ 2<x'^ 6 3 . 1,335 



1,335 
+ -^ = 8,673 feet, 
o 



3) If a string 30 feet long and weighing 8 pounds is stretched as nearly 
horizontal as possible by a force of 20 pounds, the vertical tension is 
F = |- G^ = 4 pounds, and the horizontal force 

H^ V^"^^F"=:V202-42 = V384 = 19,596 pounds, 
the tangent of the angle of inclination at the point of suspension is 

Y 4 



tang. </> = ^ 



= 0,20412, 



H 19,596 
and the angle itself is 11° 32' ; the measure of the horizontal tension is 

c = - = ^: |. = ^ H=^ 73,485 feet, 

y oU o 

the width of the span is 

3J=3{l -1 . (1)']= 30 . [1 - i- . ( Jg)]=30-0,208 = 29,792fl., 

and the height of the arc 

/"S / 3 29,792 . 0,208 , -^ 

a = y --h(l-l)=y ^ ^ ~ = V 29,792 . 0,078 = 1,524 feet. 

Kemark 1. — We find from the radius (7^= C B = CB = r and the 
ordinate ^ if = 2/ of an arc of a circle A B, Fig. 243, the ordinate 
A N = B N=y^oi half the arc A D =z B D^hj putting 

T^ = TW-" + BW^ = AM^ + (CB - G My 

= AM^+iGB- ^0~A}-AM^)^ = 2 CA^-2 GA ^GA^ — AW, 

Fig. 243. 




I.E. 4 2/i^ = 3r2 — 2r ^/r^ — y^. 
Hence we have 






V 



y 



2 

small compared with r, 



, or approximatively, if y is, 



By repeated application of this formula we find the 
ordinate of a quarter of the arc 

- 2 \^ + 8 r^/ - 4 V + 8 W \ ^ ^ • 8rV 



^2 



and that of an eighth of the arc 



^»- 2 \ ^ SrV 8\ 



= ^ 1 + 



11 
8rs 



)(i + i 



y (t + ay /i) 



( 



1 + [1 + I- + 



(D^lgT-s/- 



Since the ordinates of very small arcs can be put equal to the arcs 
themselves, we obtain for the arc A B approximatively 



161.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



299 



s = 8 . 2/3 = y (l + [1 + i + (i)^] ^)> or more accurately 

. = y (1 + [1 + i + a)^ + (i)^ + . . .] 87-.). 

But 1 + i + (i)3 + (i)3 + . . . is = r :=! (see Ingenieur, page 83), 

and tlierefore 

or substituting instead of r the abscissa B M = x by putting 3 t" a; — y^y 
we obtain 

2" 



=rif^ 



(5)] 



2/. 



This formula is not only applicable to the arc of a circle, but also to all 
low arcs of curves. 

Remark 3. If we compare the equation 



Fig. 244. 



y 







found above, with the equation of the ellipse 



y =-V2ax — x'' 



a 



(see Ingenieur, page 169), we find 

— = c and -— = i, and consequently 



a = S c and d = a v'^- = c VS. 
The curve formed by a powerfully stretched string can therefore be 
considered as the arc A CB, Fig. 344, of an ellipse, the major axis of which 

isKG=a = dc and the minor axis is K I) = K E = 5 = <;V3 = 
a Vi = 0,577 a. 

(§ 161.) Equation of the Catenary. — The complete equa- 
tion of the catenary can be found in the following manner by the 
aid of the calculus. According to § 158, we have for the angle of 

suspension T N =■ (p. Fig. 245, 
formed by the tangent 2" to a 
point of the catenary A C B with 
the horizontal co-ordinate N, 
when the arc C is denoted by s 
and the horizontal tension hy H = 

tang. 4> =- 

But is also equal to the angle 
P R formed by the element of 



Fig. 345. 

M 



B 



\^ 


Q / 


\ 


y 


N / 





r 






300 GENERAL PRINCIPLES OF MECHANICS. [§ 163. 

the SLYC P — d s with the element P B = d y of the ordinate 
N = y, and hence 

in which i? is considered as an element d x of the abscissa C N 

= X. From the above it follows, that 

dx _s dy" _ & 

dy c' dx" ~ s^' 

But d s"" is = d x* + d y\ or d y"" = d s^ — d x% whence 

d s"" — d x'' _ c"" 

dx' "~ s^' 

Clearing the equation of fractions and transposing, we obtain 

sds 
d x^ (s' + &) = s" ds^,OY dx= ,/— „ „ . 

^ ' V s -h c 

Putting s^ + c" = u, we have 

2 s ds = du and dx = - — r- = ? w^ d u. 

By integration we obtain (according to Article 18 of the In- 
troduction to the Calculus) 

a? = i / u—^du= i . -y" + Const = Vu -\- Const. 

= Vs' + 0' + Const 

OPinally, since for a; = 0, s is also = 0, we have = V^ -h Const, 
I.E. Const = — c and 

1) x= V s^ -\- c^ — c, or inversely 

s = V {x + cf — c' = Vii, c X + x", and 

_ s^ — x^ 

Example. — If a chain A G B,10 feet long and weighing 30 pounds, is 
suspended in such a manner that the height of the arc is G M = 4: feet, we 
have 

y = 1^ = 3 pounds, 

S3 _ 2,2 ^ 58 ^ 42 ^ 

"^ "" 2x "" 8 ~ ^' 
and consequently the horizontal tension 

J?=C7 = 3.f = 3| pounds. 

(§ 162.) As in the last paragraph by eliminating d ywe obtained 
an equation between the arc 5 and the abscissa x, in like manner 
by eliminating d xwe can deduce an equation between the arc 5 
and the ordinate y. Tor this purpose we substitute in the equation 



§163.] EQUILIBRIUM IN FUNICULAR MACHINES. 301 

g^, = -^,dx' = ds''-df 

and obtain the equation 
s' d s" — dy 



e df 

cds 



, or d y" (s' + c") — & d s\ whence 



Dividing the numerator and denominator by c and putting 



^ = v,we obtain 



^<l) 



, -, c d V 

dy = — , - : = , 

and the formula XIII, in Article 26 of the Introduction to the 
Calculus, gives us the corresponding integral 

f dv 

2)y = cl[ y 

Substituting in this formula s = V2cx -\-x', we obtain the 
proper equation for the co-ordinates of the common catenary 

ox .fc-\-x+ V2cx-{-x\ 
S) y = cl [ ), or 

A\ — 7 ( ^ + ^ \ — ^' ~ ^'^ 7 /l1l?\ 
^ ^ ~ \s — xl ~ 2 X \s — x; 

Finally, by inverting 2 and 3, we obtain 
5) s= (i_ -TJ.^and 



6)x=[i(-^,-f)-l]., 

e denoting the base 2,71828 ... of the Naperian system of loga- 
rithms (see Article 19 of the Introduction to the Calculus). 

Example. — The two corresponding co-ordinates of a point of the cate- 
nary are a; = 3 and y = 3 ; required the horizontal tension c of tins curve. 
Approximatively; according to No. 3 of paragraph 160, we have 

y^ X 9 2 ^^^ 



302 GENERAL PRINCIPLES OF MECHANICS. [§163. 

But according to No. 3 of this paragraph (162), we have exactly 

-.{c + X + V 2 c a? + x^\ 
y = cl y ^ j , I.E. 



,/c + 2 + V4c + 4\ 
3 = cZ( -c } 

Substituting for c, 2,58, we find the error 

, /4,58 + 2 V 3,58\ /8,3642\ 

/ = 3 - 2,58 I ( • a.58 ) = ^ - ^'^« ^ fe-) 

= 3 — 3,035 = — 0,035. 
If, however, we assume c = 2,53, we find the error 

= 3 - 3,002 = — 0,002. 
In order to find the true value of c, we put according to a well known 
rule (see Ingenieur, page 76) 

c — 2,58 / 0,035 
c - 2,53 /i ~ 0,002 - ^''''' 
whence it follows that 16,5 . c = 17,5 . 2,53 — 2,58 = 41,69 and 

41,69 

e = ^77^- = 2,527 feet. 
16,5 

Remake. — We can express very simply s, x and y for the common cate- 
nary in terms of the angle of suspension ^ ; for from what precedes we have 

c sin, <t> 
g = c tang, (b = 

COS. (p 



. ,- 5— .. C (1 — COS. (f) 

x = ciy \ + tang.'' (p — 1) = -^ ^ 



COS. (p 



and 



y = cl (tang. ^ + V 1 + tang.^ (f) = cl \ — |. 

By means of these formulas we can easily calculate the lengths of the 
arcs and of the co-ordinates for different angles of suspension, and a useful 
table, such as is given in the Ingenieur, page 353, may be thus prepared. 
For this purpose we need adopt as base but a single catenary, and in this 
case the best one is that, in which the measure of the horizontal tension is 
= 1 ; to obtain s, x and y for another catenary corresponding to the hori- 
zontal tension c, we have but to multiply the values of s, x and y given in 

s y 

the taoie by c. If tang. ^ were not = -, but to -, we would have the com- 

C G 

mon parabola, for which 

c r sin. (p ^ ^ /i ^ + </>\"l 



c „ c (sin. (py _ 
a? = -X tang.^ 6 = - I J and 

** ^ ^ 2 \cos. <pj 



fnmn ^ A ^^ — \ 

c sin. (p 



y=c tang, cp = 



cos. <p 



§ 163.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



303 



§ 163. Equilibrium of the Pulley. — Eopes, belts, etc., are 
the ordinary means employed to transmit forces to the pulley and 
the wheel and axle. We will here discuss only the most general part 
of the theory of these two apparatuses, so far as it can be done with- 
out taking into consideration the friction and the rigidity of cordage. 

A pulley (Fr. poulie ; Ger. Eolle) is a circular disc or sheave 
ABC, Figs. 246 and 247, movable about an axis and around 

Fig. 246. Fig. 247. 





whose circumference a string is laid, the extremities of which are 
pulled by the forces P and Q. The block (Fr. chape ; Ger. Gehause 
or Lager) of a fixed pulley (Fr. p. fixe ; Ger. feste R), in which the 
axles or journals rest, is immovable. That of a movable pulley 
(Fr. p. mobile ; Ger. lose E.) on the contrary is free to move. 

When a pulley is in equilibrium, the forces P and Q at the ex- 
tremities of the cord are equal to each other ; for every pulley is a 
lever with equal arms, which we obtain by letting fall from the 
axis C the perpendiculars G A and C B upon the directions D P 
and D Q of the forces or cords. It is also evident, that during any 
rotation about C the forces P and Q describe equal spaces r 13, 
when r denotes the radius CA = G B and i3° the angle of rotation, 
and from this we can conclude, that P and Q are equal. The forces 
P and Q give rise to a resultant G R ^^ R, which is counteracted 
by the journal or axle and is dependent upon the angle AD B — a 
formed by the directions of the cords, it is given by construction 
as the diagonal of the rhomb G P^R Qx constructed with P and a ; 



its value is 



a 



R = ^ P COS. ^. 



304 



GENERAL PRINCIPLES OF MECHANICS. 



[g 164. 



§ 164. The weight to be raised or the resistance Q to be overcome 
in a fixed pulley, Fig. 246, acts exactly in the same manner as the 
force P, and the force is therefore equal to the resistance, and 
the use of this pulley produces no other effect than a change of 
direction. 

On the contrary, in a movable pulley, Fig. 247, the weight E 
acts on the hook-shaped end of the bearings of the axle, while one 
end of the rope is made fast to some immovable object ; here the 
force is 

R 



P = 



2 COS. 



a 

2 



Designating the chord A M B corresponding to the arc covered 
by the string by a and the radius C A = O B, as before, by r, we 
have 



a 



a = 2 A M=2CA cos, C A M=2CAcos.ABM= 2rcos.^ 
and therefore 



r 
a 



2 cos, ^ 



and -75- 
a K 



T 

a 



Fig. 248. 



Hence, in a movable pulley, the force is to the load as the 
radius of the pulley is to the chord of the arc covered by the string. 
\i a =. % r, I.B. if the string covers a semicircle. Fig. 248, the 
force is a minimum and is P = ^ i^ ; if a = r or 
if 60" of the pulley is covered by the string, we 
have P — R. The smaller a becomes, the greater 
is P ; I.E., when the arc covered by the cord is 
infinitely small, the force P is infinitely great. 
The relation is inverted, when we consider the 
spaces defcsribed ; if 5 is the space described by 
P, while R describes the space h, we have Ps — 
Rh, whence 

s _ a 
}b~ r 

The movable pulley is a means of changing 
the force, and is used to gain power ; by means 
of it we can, e.g., raise a given load with a smaller 
force ; but in the same ratio as the force is in- 
creased the space described is diminished. 




§ 165.] 



EQUILIBRIUM IN FUNICULAR MACHINES. 



305 



Remabk. — The combinations of pulleys, such as block and tackle, etc., 
as well as the influence of friction and of the rigidity of cordage upon the 
state of equilibrium of pulleys, will be treated in the third volume. 



Fig. 249. 




§ 165. Wheel and Axle. — The wheel and axle (Fr. roue sur 
Tarbre, Ger. Eadwelle) is a rigid combination A B F E, Fig. 249, 

of two pulleys or wheels mov- 
able about a common axis. 
The smaller of these wheels 
is called the axle (Fr. arbre, 
Ger. Welle), and the larger 
the wheel (Fr.roue, Ger. Ead). 
The rounded ends E and F, 
upon which the apparatus 
rests, are called the journals 
(Fr. tourillons, Ger. Zapfen). 
The axis of revolution of a 
wheel and axle is either hori- 
zontal, vertical or inclined. 
We will now discuss only 
the wheel and axle, movable 
around a horizontal axis. We 
will also suppose, that the forces P and Q or the force P and the 
weight Q act at the ends of perfectly flexible ropes, which are 
wound around the circumferences of the wheel and of the axle. 
The questions to be answered are, what is the relation between the 
force P and the weight ft and what is the pressure upon the bear- 
ings at E and F ? 

If at the point C, where the plane of rotation of the force P 
cuts the axis E F, we imagine two equal opposite forces C P = P 

and C P =^ — P to be acting in a direction parallel to that of the 
force of rotation P, we obtain by the combination of these three 
forces a force C P = P, which acts upon the axis, and a couple 

(P, — P), whose moment i^ = P . G A = P a, when a designates 
the arm of the force A P = P or the radius A of the wheel. 

Now if we imagine the two forces D Q — Q and P Q = — Q tohe 
applied at the point D, where the plane of revolution of the weight 
Q cuts the axis E F, we obtain also a force D Q — Q acting upon 

the axis and a couple (ft — §), whose moment is — Q . B B= Qh, 
"when h designates the arm of the weight Q applied in B or the 
20 



306 GENERAL PRINCIPLES OF MECHANICS. [§166. 

radius D B of the axle. Since the axial forces C P = P and 
B Q — Q are counteracted by the bearings, and consequently 
can have no influence upon the revolution of the machine, it is 
necessary, in order to have a state of equilibrium, that the two 
couples, which act in parallel planes, shall have equal moments 
(compare § 94), or that 

P a = Q d, or 

P _ b 

Q~ a 

In every wheel and axle tuMch is in equilibrium, whatever may 
te its length, the moment P a of the power is, as in the lever, equal to 
the moment Qb of the load, or the ratio of the poiver to the load is 
equal to that of the arm of the load to the arm of the poioer. 

If more than two forces act upon the wheel and axle, the sum 
of moments of the forces tending to turn it in one direction is 
naturally equal to the sum of those tending to turn it the other. 

§ 166. The axial forces C P = P and i) § = § can be 
decomposed into the vertical forces C Pi = P^ and D Q^ = Qi and 
into the horizontal forces C P^^ — P. 2 and D Q^— Q^; the first two 
forces combined with the weight of the machine G, which acts at 
the centre of gravity S of the machine, give the total vertical 
pressure on the bearings, which is 

r, + V, = p,+ Q,+ G, 

while the horizontal forces P.> and Q.> produce the lateral pressures 
Hi and II^ on the bearings. If a is the angle of inclination P P<i 
of the direction of the force P to the horizon and /3 that Q P Qc^ 
of the load, we have 

P^ — P sin. a and P2 — P cos. a, as well as 

Q^= Q sin, (3 and Q^ — Q cos. [3. 



If now I is the total length of the axis E F, d the distance E, 

e the distance D E and c the distance 8 E of the points of the axis 
C, D and *S' from one extremity ^ of the axis, we have, according to 
the theory of the lever: 

1) When we consider E as fulcrum of the lever E F, which is 
acted on by the forces Pj, Qi and G, 



V, , EF= P,.E G + Q,. ED + G.E S, i.e. 

V^l = Pid + Q^e -{- G s, 



166. 



EQUILIBRIUM IN FUNICULAR MACHINES. 



307 



whence we obtain the yertical pressure 



F. - 



I 



, and 



2) considering F as the fulcrum of the supposed lever, 

Fi . FE = P, . ¥C + Q, . FD + G . FS, LE. 
Vrl= F,{1- d) + Q, (l — e) + G{1- s), 
whence we deduce the vertical pressure 

P, {l-d) + Q, {I -e) + G{1- s) 



Fi- 



I 



Fig. 249. 




The horizontal pressures J/j and H^ are found, as follows, from 
the horizontal forces P.. and Q.,. 

1) Considering F as the fulcrum of the lever F F acted on by 
the forces Ps and Q.2, we obtain 

m . FF = P, . FlJ - Q, . FD, I.E. 
H^l = P^ d — Q.2 e, 
whence we obtain the horizontal pressure 

P, d — Q^e 



ff,,= 



I 



2) Considering F as the fulcrum, we have 

ff, . Vf -= P, . FT' - Q, . FF, I.E. 
ir,I= P,{1- d) - Qo{l- e), 
from which we deduce the horizontal pressure 
^ P, (I - d) - Q.2 (I - e) 

H — , 



308 GENERAL PRINCIPLES OF MECHANICS. " [§ 166. 

By the application of the parallelogram of forces, we obtain the 
total pressures Ri and i?2 upon the bearings E and F, and they are 
R, =z VV,' + ^ and R, ^ VV,' + H~\ 

Finally, if 6^ and 6^ are the angles Ri E H^ and R^ F H^ formed 
by these pressures with the horizon, we have 

tang, d^ = -~ and tang. 6^ — — ?. 

Example. — The weight Q, suspended to a wheel and axle, acts verti- 
cally and weighs 365 pounds ; the radius of the wheel is a = If feet ; the 
radius of the axle is ?> = f foot ; the weight of the wheel and axle together 
is 200 pounds ; the distance of its centre of gravity from the journal E is 
1^ feet ; the centre of the wheel is at a distance d = ^ from this journal 
E^ and the vertical plane, in which the weight acts, is e = 3 feet distant 
from the same point, while the whole length of the axis is E E = I = 4: 
feet ; now if the force necessary to produce equilibrium acts downwards at 
an angle of inclination to the horizon of a = 50°, how great must it be and 
what are the pressures upon the bearings ? Here we have Q = 365, /3 = 
90°, and consequently Q^ z= Q sin. (3 = Q and Q2 = Q cos. p = 0, P is 
unknown, and a is = 50°, whence P^ = P sin. a = 0,7660 . P and P^ 
Is = P COS. a = 0,6428 , P, but a is = If = ^ and & = |, whence 

b 
P = -Q = f.dQ5 = 156,4 pounds, P^ = 119,8 and P^ = 100,5 pounds. 

Since 1 = 4, d = ^, e = 2 and s = f , we have I — d = ^:^-, I — e = 2 
and Z — « — f . 

1) On the bearing E the vertical pressure is 

119,8. f + 365 . 2 +200. f ^^^ ^ 
F3 = ^-— — J ~ — — - = 280,0 pounds, 

and the horizontal pressure is 

^ 100,5.f-0.2 

//g = — ^ = 18,8 pounds, 

and consequently the resulting pressure is 



Pg = i/V^^ + B^^ = 1/2802 + 18,82 = 280,6 pounds, 

and its inclination to the horizon is determined by the formula 

280 
tang. 6^ = ' , log tang. 6^ = 1,17300, from which we obtain 6^ = 86° 9' 5". 

18,8 

2) For the bearing at E 

119,8 . J^ + 365 . 2 + 200 . I .... . , 

Fi = J = 404,8 pounds and 

^ 100,5 .^ - 

H^ = — - — -^ = 81,7 pounds, 

and consequently the resulting pressure is 



M^ = i/ Fi2 + H^2 =z |/404,82 + 81,72 =r 413,0 pounds,. 



§ 167.] RESISTANCE OF FRICTION, ETC. 309 

and for its inclination d^ to the horizon we have 

404,8 , „ . 

^^^- ^1 = -^tV' % ^^^9' ^1 = 0,69502 or 6, = 78° 35'. 
ol,7 

We see that these results are correct, for we have 

V^ +Vz = 380 + 404,8 = 684,8 = P^ + Q^ + G, and 

E^+H^= 81,7 + 18,8 = 100,5 == Pg + Q^. 



CH APT ER V. 

THE RESISTANCE OF FRICTION AND THE RIGIDITY OF CORDAGE. 

§ 167. Resistance of Friction. — Heretofore we have sup- 
posed (§ 138) that two bodies could act upon one another only by 
forces perpendicular to their common plane of contact. If these 
bodies w^ere perfectly rigid and their surfaces of contact mathemat- 
ical planes, i.e. unbroken by the smallest hills or hollows, this law 
would also be confirmed by experiment ; but since every material 
body possesses a certain degree of elasticity or even of softness, and 
since the surface of all bodies, even the most highly polished, con- 
tains small hills and valleys and in consequence of the porosity of 
matter does not form a perfectly continuous plane, when two bodies 
press upon each other their points of contact penetrate, pro- 
ducing an adhesion of the parts, which can only be overcome by a 
particular force, whose direction is that of the plane of contact. 
This adhesion of bodies in contact, produced by their mutual pene- 
tration and grasping of each other, is what is called friction (Fr. 
frottement, Ger. Reibung). Eriction presents itself in the motion 
of a body as a passive force or resistance, since it can only hinder 
or prevent motion, but can never produce or aid it. In investig^i- 
tions in mechanics it can be considered as a force acting in opposi- 
tion to every motion, whose direction lies in the plane of contact 
of the two bodies. Whatever the direction -may be in which we 
move a body resting upon a horizontal or inclined plane, the fric- 
tion wdll always act in the opposite direction to that of the motion, 
E.G., when we slide the body down an inclined plane, it will appear 
as motion up the same. If a system of forces is in equilibrium, the 
smallest additional force produces motion as long as the friction 
does not come into play ; but when friction is called into existence 
a greater addition of force, the amount of which depends upon the 
friction, is necessary to disturb the equilibrium. 



310 GENERAL PRINCIPLES OF MECHANICS. [§ 168, 169. 

§ 168. In overcoming the friction, the parts which come in 
contact are compressed, the projecting parts bent oyer, or perhaps 
torn away, broken off, etc. The friction is therefore dependent not 
only upon the roughness or smoothness of the surfaces, but also 
upon the nature of the material of which the bodies are composed. 

The harder metals generally cause less friction than the softer 
ones. We cannot establish a ^^riori any general rules for the de- 
pendence of friction upon the natural properties of bodies ; it is in 
fact necessary to make experiments upon friction with different 
materials, in order to be able to determine the friction existing 
between bodies under other circumstances. The unguents (Fr. les 
enduits ; Ger. die Schmieren) have a great influence upon the 
friction and upon the wearing away of bodies in contact. The 
pores of the bodies are filled and the other roughnesses diminished 
by the fluid or half fluid unguents, such as oil, tallow, fat, soaps, 
etc., and the mutual penetration of the bodies much diminished ; 
for this reason they diminish yery considerably the friction. 

But we must not confound friction with adhesion, i.e., with 
that union of two bodies which takes place when the bodies come 
in contact in yery many points without the existence of any pres- 
sure between them. The adhesion increases with the surface of 
contact and is independent of the pressure, while for friction the 
reverse is true. When the pressures are small, the adhesion appears 
to be yery great compared with the friction, but if the pressures 
are great, it becomes but a very small portion of the friction and 
can generally be neglected. Unguents generally increase the adhe- 
sions, since they produce a greater number of points of contact. 

§ 169. Kinds of Friction. — We distinguish two kinds of 
friction, viz., sliding and rolling friction. The sliding friction 
(Fr. frottement de glissement ; Ger. gleitende Keibung) is that 
resistance of friction produced, when a body slides, i.e., moves so . 
that all its points describe parallel lines. Rolling friction (Fr. f. de 
roulement ; Ger. rollende or walzende Reibung) on the contrary, 
is that resistance developed, when a body rolls, i.e., when every 
point of the body at the same time progresses and revolves and 
when the point of contact describes the same space upon the 
moving body as upon the immovable one. A body M, Fig. 250, 
supported on the plane H R, slides, for example, upon the plane 
and must overcome sliding friction, when all points such as A, B, C, 
etc., describe the parallel trajectories A A^, B B^, O C„ etc., and 
therefore the same points of the moving body come in contact with 



§ 170.] 



RESISTANCE OF FRICTION, ETC. 



311 



different ones of the support. The body M, Fig. 251, rolls upon 
the plane H R and must therefore overcome rolling friction, when 



Fia. 250. 



Fm. 251. 





the points A, B, etc., of its surface move in such a manner, that 
the space A E B^ = the space A D B — A^ Di B^ and also that 
space ^ ^ is = the space A D and the space Bx E — B^ D^, etc. 

A particular kind of friction is the friction of axles or journals 
which is produced, when a cylindrical axle, journal or gudgeon 
revolves in its bearing. We distinguish two kinds of axles, hori- 
zontal and vertical. The horizontal axle, journal or gudgeon 
(Fr. tourillon ; Ger. liegende Zapfen) moves in such a manner that 
different points of the gudgeon, etc., come successively in contact 
with the same point of the support. The vertical axle or pivot 
(Fr. pivot; Ger. stehende Zapfen) presses with its circular base 
upon the step, on which the different points of it revolve in con- 
centric circles. 

Particular kinds of friction are produced, when a body oscillates 
upon an edge, as, e.g., a balance, or when a vibrating body is sup- 
ported upon a point, as, e.g., the needle of a compass. 

Friction can also be divided into immediate (Fr. immediat ; 
Ger. unmittelbare) and mediate (Fr. mediat ; Ger. mittelbare). In 
the first case the bodies are in immediate contact ; in the latter, 
on the contrary, they are separated by unguents, as, e.g., a thin 
layer of oil. 

We distinguish also the friction of repose or quiescence (Fr. f. de 
repos ; Ger. E. der Euhe), which must be overcome when a body 
at rest is put in motion, from the friction of motion (Fr. f de 
mouvement ; Ger. E. der Bewegung), which resists the continuance 
of a motion. 

§ 170. Laws of Frictions. — 1. The friction is proportional 
to the normal pressure between the rubbing bodies. If we press 
a body twice as much against its support as before, the friction 
becomes double. A triple pressure gives a triple friction, etc. 
If this law varies slightly for small pressures, we must ascribe the 
variations to the proportionally greater influence of the adhesion. 



312 



GENERAL PKINCIPLES OF MECHANICS. 



[§ 171. 



2. The friction is independent of the rubbing surfaces or sur- 
faces of contact. The greater the rubbing surfaces the greater is, 
it is true, the number of the rubbing parts, but the pressure upon 
each part is so much the smaller, and consequently the resistance 
of friction upon it is less. The sum of the frictions of all the parts 
is therefore the same for a large and for a small surface, when the 
pressure and other circumstances are the same. If the surfaces of 
the sides of a parallelopipedical brick are of the same nature, the 
force necessary to move the brick on a horizontal plane is the same 
whether it lies on the smallest, medium, or greatest surface. When 
the surfaces are very great and the pressures very small, this rule 
appears to be subject to exceptions on account of the effect of the 
adhesion. 

3. The friction of quiescence is generally greater than that of 
motion, but the latter is independent of the velocity ; it is the 
same for high and low velocities. 

4 The friction of greased surfaces (mediate friction) is gene- 
rally smaller than that of ungreased surfaces (immediate friction) 
and depends less upon the rubbing bodies themselves than upon 
the unguent. 

0. The friction on axles is less than the ordinary friction of 
sliding. The rolling friction between smooth surfaces is in most 
cases so small, that we need scarcely take it into account in com- 
parison with the friction of sliding. 

Remark. — The foregoing rules are strictly true only, when the pressure 
upon the unit of surface of the bearings is a medium one, and when the 
velocity of the circumference of the journal does not exceed certain limits. 
This medium pressure is from 250 to 500 pounds per square inch, and the 
mean velocity of the circumference should be 2 to 10 inches. When the 
pressure is much smaller, the adhesion forms a very sensible portion of the 
resistance which then becomes dependent upon the magnitude of the rub- 
bing surfaces, and where the pressure and velocity are very great a large . 
quantity of heat is developed, which volatilizes the unguents, thus causing 
the journals to cut very quickly. When, as in the case of turbines, rail- 
road cars, etc., we cannot avoid these 
great velocities, we must counteract this 
heating of the axle by increasing the rub- 
bing surfaces, i.e., by increasing the length 
and thickness of the axles. 

§ 171. Co-eflBcient of Friction. 

— From the first law of the foregoing 
paragraph we can deduce the fol- 
lowing. If in the first place a body 




cor- 



§171.] RESISTANCE OF FRICTION, ETC. 313 

A Cy Fig. 252, presses with a force N against its support, and if 
to move it along, i.e., to overcome its friction, we require the 
force F, and if in the second place, when pressing with the force 
Ni a force F^ is necessary to transfer it from a state of rest into 
one of motion, we will have, according to the foregoing paragraph, 

FN F, 

-— — -^y whence F — -^ . N. 

F^ N' iv; 

If by experiment we have found for a certain pressure N^ the 
corresponding friction F^, we see from the above, that if the rub- 
bing bodies and other circumstances are the same, the friction F 
corresponding to another pressure N can he found hy multiplying 

this pressure hy the ratio (-^/ hetween the values Fx and iVj 

responding to the first observation. 

This ratio of the friction to the pressure or the friction for a 
pressure = 1, e.g. pound, is called the coefficient of friction 
(Fr. coefficient du frottement ; Ger. Eeibungscoefficient) and will 
in future be designated by 0. Hence we can put in general 

F= cf) N. 

The coefficient of friction is different for different materials 
and for different conditions of the same material and must there- 
fore be determined by experiments undertaken for that purpose. 
If the body A C is pulled along a distance s upon its support, the 
work to be performed is F s. The mechanical effect N s ab- 
sorbed by the friction is equal to the product of the coefficient of 
friction, the normal pressure and the space described. If the sup- 
port is also movable, we must understand by 5 = Si — 53 the relative 
space described by the body, and F s — </> i\^ 5 is the work done by 
the friction between the two bodies. The body that moves the 
most quickly must perform, while describing the space ^i, the me- 
chanical effect (f) N s-i and the body which moves slower gains in 
consequence of the friction while describing the space So the me- 
chanical effect iVsa ; the loss of mechanical effect caused by the 
friction between the two bodies is 

(p N Si — (t> N Si = N (si — So) = Ns. 

Examples — 1. If for a pressure of 260 pounds the friction is 91 pounds, 
the corresponding coefficient of friction is.^ = -^q = ^ = 0,35. 

2. In order to pull forward a sled weighing 500 pounds on a horizontal 
and very smooth snow-covered road, when the coefficient of friction is 
(p = 0,04, a force F = 0,04 . 500 = 20 pounds is necessary. 

3. If the coefficient of friction of a sled loaded with 500 pounds and 



314 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 172. 



pulled over a paved road is 0,45, the mechanical effect required to move 
the sled 480 feet is <^ JV's = 0,45 . 500 . 480 = 108000 foot-pounds. 



Fig. 253. 




172. The Angle of Friction or of Repose and the 

Cone of Friction.— If a body 
A C, rig. 253, lies upon an in- 
clined plane F H, whose angle of 
inclination is F H B = a, we can 
decompose its weight into the nor- 
mal pressure N = G cos. a, and 
into the force S = G sin. a paral- 
lel to the plane. The first force 
causes the friction F — (h G cos. a, 
which resists every motion upon 
the plane ; consequently the force necessary to push the body up 
the plane is 

P — F ■\- 8 — (^ G COS. a ^ G sin. a 
— {si7i. a -\- ([) cos. a) G, 
and the force necessary to push it down the same is 
P^ -- F — S = {(p cos. a — sin. a) G. 
The latter force becomes = 0, i.e. the body holds itself upon 
the inclined plane by its friction when siJi. a = (p cos. a, i.e. when 
tang, a ~ 0. As long as the inclined plane has an angle of incli- 
nation, whose tangent is less than (p, so long will the body remain 
at rest upon the inclined plane ; but if the tangent of the angle of 
inclination is a little greater than (p, the body will slide down the in- 
clined plane. We call this angle, i.e. the one whose tangent is equal 
to the coefficient of friction, the angle of friction or of repose or of 
resistance (Fr. angle du frottement, Ger. Eeibungs— or Euhewinkel). 
Hence we obtain the coefficient of friction (for the friction of qui- 
escence) by observing the angle of friction p and jDutting (p = tang. p. 
In consequence of the friction, the surface F II, Fig. 254, of a 
body counteracts not only the normal pressure iV^ of another body 

A B, but also any oblique pressure P when 
the angle N B P = a formed by its direc- 
tion with the normal to the surface does 
not exceed the angle of friction ; since the 
force P gives rise to the normal pressure 

B N — P COS. a, and to the lateral or 




a 



tangential pressure B 8 = S = P sin 
and since the normal pressure P cos. a pro- 
duces the friction (p P cos. a, which opposes 



§ 173.] 



RESISTANCE OF FRICTION, ETC. 



315 



every movement in the plane F H, 8 can produce no motion as 
long as we have 

</) P COS. a'^ P sin. aoY(f) cos, a > sin. a, i.e. 
tang, a < or a < p. 
If we cause the angle of friction C B D = p to revolve about 
the normal C B, it describes a cone, which we call the cone of fric- 
tion or of resistance (Fr. cone de fr., Ger. Reibungskegel). The 
cone of friction embraces the directions of all the forces, which are 
completely counteracted by the inclined plane. 

Example. — In order to draw a full bucket weighing 200 pounds up a 
wooden plane inclined to ttie horizon at an angle of 50°, the coefficient of 
friction being 9 = 0,48, we would require a force 

P = (0 c(?s. a + sin. a) G = (0,48 co8. 50° + sin. 50°) . 200 
= (0,308 + 0,766) . 200 = 215 pounds. 
In order to let it down or to prevent its sliding down, we would have need 
of a force 

Pj = (6 CDS. a — sin. a) G = — (sin. 50° — 0,48 cos. 50°) . 200 
= - (0,766 — 0,308) . 200 = — 91,5 pounds. 

§ 173. Experiments en Friction. — Experiments on friction 

have been made by many persons ; those, Avliich were most ex- 

. tended and upon the largest scale, are the experiments of Coulomb 

,; and Morin. Both these experimenters employed, for the determina- 

ftion of the coefficient of friction of sliding, a sled movable upon a 

" horizontal surface and dragged along by a rope passing over a fixed 

pulle}', to the end of which a weight was attached, as is shown in 

Fig. 255, in which ^ ^ is the surface, C D the sled, E the pulley, 

and F the weight. In order to obtain the coefficients of frictions 

for different substances, not only the runners of the sled, but also 

the surface upon which it slid, were covered with the smoothest 

possible plates of the material to be experimented on, such as wood, 

iron, etc. The coefficients of friction of rest were given by the 

weight necessary to bring 
the sled from a state of 
rest into motion, and the 
coefficients of friction of 
motion were determined 
by aid of the time required 
by the sled to describe a 
certain space s. If G is 
the weight of the sled and 



P the weight necessary to move the same, we have the friction 




316 GENERAL PRINCIPLES OF MECHANICS. [§173 

P ' C 
=: (f) G, the moving force = F — (j) G and tlie mass M = — - — ' 

g 
whence, according to § 68, the acceleration of the uniformly acceler- 
ated motion engendered is _ F — (j) G 

^ '~ F + G ^' 
and inversely the coefficient of friction is 

_F F+ G p 

^ ~ G G ' g 

2s 
But we have also (§ 11) s = ^^P ^^ whence^ — — - and 

P _ F + G 2_s 
^~ G G ' gf 

If we allow the sled to slide down an inclined plane, the moving 

force is = 6^ {sin. a — (p cos. a), and the accelerated mass is = — ; 

consequently the acceleration is 



2 s G (sin. a — (h cos. a) , . , . 

= -^ = — ^ — = g [sm. a — <p cos. a) 

J 



^ f G 



2 s 
or —7^ = sin. a — (f) cos. a, and consequently the coefficient of sliding 

2 s 

friction is (b = tang, a . 

^ g t COS. a 

If 7i denotes the altitude, I the length and a the base of the 
inclined plane, we have also = 



a g a f 

In order to determine the coefficient of friction for the friction 
of axles or journals, they employed a fixed pulley A C B, Fig. 256, 
around which a rope was wound, to which the weights F and Q 
were suspended; from the sum of the weights F + Q we have 
the pressure E upon the axle, and from their difference F — Q the 
force at the periphery of the pulley, which is held in equilibrium by 
the friction F = (f) {F + Q) on the surface of the axle. If now 
C A = a = the radius of the axle and C D = r = the radius of the 
journal, we have, since the statical moments are equal, 

{F- Q)a=Fr = (j){F + Q)r, 
and consequently the coefficient of friction of rest 

, ^~F+ Q' r' 

and, on the contrary, when the weight F falls and Q rises in the 
time t a distance s, the coefficient of friction of motion is 



173.] 



RESISTANCE OF FRICTION, ETC. 






317 



<P+Q 9 

The engineer Hirn employed in his (the latest) experiments 
upon friction of journals the apparatus represented in Fig. 257, 
Fig. 256. Fig. 257. 





"which he called a friction balance (Fr. balance de frottement, Ger. 
Keibungswage). Here C is an axle, which is kept in constant 
rotation, as, e.g., by a water-wheel, D is the bearing, and A D B 
is a lever of equal arms, which produces the pressure between 
the journal and its bearing by means of the weights Pand Q. The 
pressure on the axle E — P -\- Q produces the friction 

F=ct>E = cp{F+Q) 
between the journal and its bearing. With this force the revolving 
shaft seeks to turn the bearing and the lever A D B, which is attached 
to it, in the direction of the arrow ; and therefore, in order to keep 
the whole in equilibrium, we must make the weight P on one side 
A so much greater than the weight Q on the other, that P — Q 
will balance the friction. But the friction F acts with the arm 
C D = r — the radius of the bearing and the difference of the 
weights P — Q with the arm C A = a, which is equal to the hori- 
zontal distance between the axis C of the shaft and the vertical 
line through the point of suspension A, and therefore we have 

Fr = ^Rr^(p{P-vQ)r=^{P-Q)a, 

and the coefficient of friction required 

P-Q 



<P = 



a 



P + Q ' r 

Remakk. — Before Coulomb, Amontons, Camus, Bulflinger, Muschen- 
brock, Ferguson, Ymce and others had studied the subject of fiiction and 
made experiments upon it. The results of all these researches have, however, 
little practical value; for the experiments were made upon too small a scale. 
The Bame objection applies to those of Ximenes, which were made about 



318 GENERAL PRINCIPLES OF MECHANICS. [§174. 

the same time as those of Coulomb. The results of Ximenes are to be found 
in the work " Teoria e Pratica delle resistenze de' solidi ne' loro attriti, 
Pisa, 1782." Coulomb's experiments are described in detail in the work i 
"Theorie des machines simples, etc., par Coulomb. Nouv. edit., 1821." 
An abstract from it is to be found in the prize essay of Mettemich, " Vom 
Widerstande der Reibung, Frankfurt und Mainz, 1789." The later experi- 
ments on friction were made by Rennie and Morin. Rennie emjjloyed in 
his experiments in some cases a sled, which slid upon a horizontal surface, 
^nd in others an inclined plane, down which he caused the bodies to slide, 
and from the angle of inclination determined the amount of the friction. 
Rennie's experiments were made with most of the substances, which we 
meet with in practice, such as ice, cloth, leather, wood, stone and the 
metals ; they also give important data in relation to the manner in which 
bodies wear, but the apparatus and the manner of conducting these experi- 
ments do not allow us to hope for as great accuracy as Morin seems to have 
attained in his experiments. A German translation of Rennie's Experiments 
is to be found in the 17th volume (1832) of the Wiener Jahrbiicher des 
K. K. Polytechnischen Institutes, and also in the 34th volume (1829) of 
Dingler's Polytechnisches Journal. The most extensive experiments and 
those, which probably give the most accurate results, are those made by 
Morin, although it cannot be denied that they leave certain points doubtful 
and uncertain, and that here and there there are points, upon which more 
information could be desired. This is not the place to describe the method 
and apparatus employed in these experiments ; we can only refer to Morin's 
writings : " Nouvelles Experiences sur le frottement," etc. A capital discus- 
sion of the subject " friction," and a rather full description of almost all the 
experiments upon it, Morin's included, is given by Brix in the transactions 
of the Society for the Advancement of Industry in Prussia, 16th and 17th 
Jahrgang — Berlin, 1837 and 1838. Later experiments on mediate friction, 
with particular reference to the different unguents, made by M. C. Ad. 
Hirn, are described in the " Bulletin de la societe industrielle de Mulhouse, 
Nos. 128 and 129, 1855," under the title of " Etudes sur les principaux 
phenomenes que presentent les frottements mediats, etc. ;" an abstract of it 
is to be found in the " Polytechnisches Centralblatt, 1855. Lieferung, 10.'* 
The latest researches upon friction by Bochet are described under the title, 
" Nouv. Recherches experimentales sur le frottement de glissement, par M. 
Bochet," in the Annales des Mines, Cinq. Serie, Tome XIX., Paris, 1861. 
Prof. Riihlmann gives some information in regard to the experiments with 
Waltjen's friction balance in the "Polytechnisches Centralblatt, 1861. 
Heft 10." 

§ 174. Friction Tables. — The following tables contain a con- 
densed summary of the coefi&cients of friction of the substances, 
most gsnerally employed in practice. 



§ l'?4.] 



RESISTANCE OF FRICTION, ETC. 



319 



TABLE I. 



COEFFICIENTS OF FRICTION OF REST. 



- 


Condition of the surfaces and nature of the unguents. 


Name of the rubbing bodies. 


0,30 


5j 

.a 
■| 

13 
<u 

a 

(L) 

o 

0,65 


1) 

"1 


re 
m 


c-i 

0,14 


o 

0,22 


re 

a 
re 

•73 

<L) 
X, 

tf) 

'o 
fin 

0,30 


c 

4) 
10 

"o 
S 

Ti 
re 

re 
i> 
}-. 

O 




' Minimum value. 


Wood upon 


Mean " 


0,50 


0,68 


— 


0,21 


0,19 


0,36 


0,35 




wood .... 


Maximum " 


0,70 


0,71 




— 


0,25 


0,44 


0,40 




, _ ^ , f Minimum value. 
Metal upon j 

-» Mean 
metal . . . . j 


0,15 
0,18 





0,11 
0,12 


0,10 


0,11 




0,15 




[ Maximum " 


0,24 




0,16 












W^ood on metal 


0^60 


0^65 


0,10 


012 


0,12 




10 




Hemp in ropes. 


' Mini'm value. 


0,50 














plaits, etc., on - 


Mean " 


0,63 


0,87 














wood 


Max'm " 


0,80 
















Thick sole leath- 


r 


















er as packing 


On edge . . . 


0,43 


0,62 


0,12 












on wood or 


Flat 


0,62 


0,80 


0,13 




— 






0,27 


cast iron . . . 


. 


















Black leather 


Made of wood. 


0,47 
















straps over - 
drums . . . . 


" metal 


0,54 










. — 


0,28 


0,38 


Stone or brick 


j^ 


















upon stone or 


Mini'm value. 


0,67 
















brick, well pol- 


Max'm " 


0,75 
















ished 


, 


















Stone upon wrought ( Min. val. 
iron i Max. " 


0,43 
















0,49 
0,64 
















Pearwood upon stone 

















320 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 175. 



TABLE II. 
COEFFICIENTS OF FRICTION OF MOTION. 





Condition of the surfaces and nature of the unguents. 

1 


Name of the rubbing bodies. 












d 

f 
i. 












0,20 


1 


•o 

> 

5 


1/1 

HI 
0,06 


i 

0,06 


•4-> 


Pure wagon g 


d 

O 


O 


■ Min. value. 






0,14 


0,08 


Wood upon ^ ^^^^ , 


0,36 


0,25 




0,07 


0,07 






0,15 


0,12 


^^«^ [ Max. - 


0,48 






0,07 


0,08 




— 


0,16 


0,15 


-.r . . f Min. value. 
Metal upon 


0,15 


— 


0,06 


0,07 


0,07 


0,06 


0,12 




0,11 


, 1 -{ Mean " 
metal . . . . ' 

[ Max. " 


0,18 


0,31 


0,07 


0,09 


0,09 


0,08 


0,15 


0,20 


0,13 


0,24 




0,08 


0,11 


0,11 


0,09 


0,17 


— 


0,17 


Wood uponT^-^^l"^- 
metal ^ ^^^^ " 


0,20 




0,05 


0,07 


0,06 








0,10 


0,42 


0,24 


0,06 


0,07 


0,08 


0,08 


0,10 


0,20 


0,14 


[ Max. " 


0,62 




0,08 


0,08 


0,10 


— 


— 


— 


0,16 


Hemp in ropes, ( On wood, 
etc ( On iron . 


0,45 


0,33 




















0,15 




0,19 










Sole leather flat 


' Raw . . . 


0,54 


0,36 


0,16 


— 


0,20 










upon wood or - 


Pounded. 


0,30 


















metal. .... 


Greasy . . 




0,25 
















The same on 
edge for pis- < 


Dry ... 

Greasy. . 


0,34 


0,31 
0,24 


0,14 


— 


0,14 










ton packing. 


i- 





















Remark. — More complete tables of the coeflScients of friction are to be 
found in the " Ingenieur," page 403, etc. The coeflScients of friction of 
loose granular masses will be given in the second volume, when the theory 
of the pressure of earth is treated. 

§ 175. The Latest Experiments on Friction. — From the 

experiments of Bochet upon sliding friction, we find, that the 
results obtained by the older experimenters Coulomb and Morin 
must undergo some important modifications. The former experi- 



§175.] RESISTANCE OF FRICTION, ETC. 321 

ments were made with railroad wagons weighing from 6 to 10 tons, 
which were caused to slide on a horizontal railroad either upon 
their wheels, which were made fast, or upon a kind of shoe (patin). 
The shoes were fastened to the frame of the wagon before, between 
and behind the wheels, and in the different series of experiments 
they were covered with soles of different materials, such as wood, 
leather, iron, etc., on which a pressure of 2, 4, 6, 10 and 15 kilograms 
per square centimetre could be produced. The wagon, thus 
transformed into a sled, was moved by a locomotive attached in 
front by means of a spring dynameter, which gave the pull or force, 
which balanced the sliding friction. In order to prevent, as much 
as possible, the resistance of the air, the wagon, which preceded the 
sled, had a greater cross-section than the latter. 

The correctness of the formula i^ = iV, according to which 
the friction F is proportional to the pressure, is proved anew by 
these experiments ; but it was found, that the co-efficient of fric- 
tion was dependent not only upon the nature and state of the rub- 
bing surfaces, but also upon other circumstances, viz. : the velocity 
of the sliding body and the specific pressure, i.e., the pressure per 
unit of surface. Bochet puts 

K — y 

^ 1 + a^; '' 

in which v denotes the velocity of sliding, k the value of </> for infi- 
nitely slow and y the value for a very rapid motion. According 
to this formula the coefficient decreases gradually from « to y as 
the velocity increases. The mean value of the coefficient a is 
= 0,3, when v is expressed in meters, and on the contrary = 0,091, 
when V is given in feet. Hence we can assume the co-efficient of 
friction to be constant only, when the velocities vary from to at 
most 1 foot and when the other circumstances remain the same. 
The co-efficients k and y are different for different materials and 
depend upon the degree of smoothness of the rubbing surfaces, 
upon the unguents, upon the Specific pressure etc. 

The co-efficient of friction k attains its maximum value for 
wood, particularly soft wood, leather and gutta-percha sliding upon 
dry and ungreased iron rails. Here we have a — 0,40 to 0,70. The 
mean value for soft wood is /c = 0,60 and for hard wood k — 0,55. 

The value k is also very different for the friction of iron upon 
iron. If the surfaces are not polished we have k = 0,25 to 0,60 ; 
and, on the contrary, for polished surfaces we have k = 0,12 to 
21 



32^ GENERAL PRINCIPLES OF MECHANICS. [§175. 

0,40o The friction of iron upon iron is not diminislied by sprink- 
ling it with water, but the friction of wood, leather and gutta-' 
percha is considerably diminished by wetting the rail. When the 
surfaces are oiled, a sinks to from 0,05 to 0,20. 

The co-efficient y is always smaller than a. When the velocities 
are great, the surfaces smooth, the unguent properly applied and 
the specific pressure a medium one, y has nearly the same value for 
all substances. 

The friction of rest is greater only in those cases where wood 
or leather slide upon wet or greased rails, and then it is twice as great. 

According to these experiments, we have 

1. for dry soft wood, when the pressure is at least 10 kilo- 
grams per square centimeter or 142 pounds per square inch, 

» = r^. - <•■'»■' 

2. for dry hard wood under the same pressure 

3. for half polished iron, dry or wet, under a pressure of more 
than 300 kilograms per square centimeter or 4267 pounds 
per square inch, 

0A5 

^ = r:^W-v -"''''' 

4. for the same either dry, under a pressure of at least 100 
kilograms per square centimeter or polished and greased 
under specific pressure of at least 20 kilograms, and also 
for resinous wood with water as unguent under the same 
pressure, 

5. for wood properly polished and rubbed with fatty water or 
fat under a pressure of at least 20 kilograms per square 
centimeter (284 pounds per square inch), 

0,10 ^ 

^ = iToTv ^ ^''^^' 

If V is given in feet, we must substitute in the denominator 
0,091 ?; instead of 0,3 1^. 

Remark. — It is very desirable that these experiments, made on so large 
a scale and giving results which differ so much from those already known, 
should be repeated. 



% 176.] 



RESISTANCE TO FRICTION, ETC. 



323 




4. § 176. Inclined Plane. — One of the most important applica- 
tions of the theory of shding friction is to the determination of the 
conditions of equilibrium of a body A C upon an inclined plane 

F H, Fig. 258. If, as in § 146, 
^^^' ^^^' F H E = ais the angle of incli- 

nation of the inclined plane and 
F Si = (3 the angle formed by* 
the direction of the force F with 
the inclined plane, we have the 
normal force due to the weight G 
iVo = G COS. a, 

the force which tends to move 
the body down the plane = S = 
G sin. a, the force iV,, with which 
the force F seeks to raise the 
body from the plane, = F sm. (3 and the force S^ with which it 
draws the body up the plane = F cos. (3. The resulting normal 

force is 

JS/'^ JV, — J\\ = G COS. a — F sin. 13, 

and consequently the friction is 

F= (p (G cos, a — F sin. (3). 

If we wish to find the force necessary to draw the body up the 
plane, the friction must be overcome, and therefore we have 

8x = S + F, I.E. F COS. 13 = G sin. a + cf) {G cos. a — F sin. [3). 

But if the force necessary to prevent the body from sliding down 
the plane is required, as the friction assists the force, we will have 

Si + F— S, I.E. P COS. [3 + ((){G cos. a— F sin. (3) = G sin. a. 

From these equations we obtain in the first case 



F = 



F = 



sin. 


a + (p COS. a 


COS. 


(3 -\- (p sin. [3 


sin. 


a — (f) COS. a 



: . G, and in the second case, 



G. 



COS. (3 — (p sin. (3 ' 

If we introduce the angle of friction or of repose p by putting 

sin. 
= tana, p = — — , we obtain 

^ ^ COS. p' 

p _ sin. a COS. p ± cos. a sin. p ^ 
COS. (3 COS. p- ± sin. (3 sin. p * ' 



324 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 176. 



or according to a well-known trigonometrical formula 

p ^ sin, {a ± p) ^, 
COS. (13 ^^ p) ' ' 

the upper signs are for the case, when motion is to be produced, and 
the lower ones, when motion is to be prevented. 
As long as we have 

„ . sin. (a — p) sin. {a -^ p) 

P > jT. ^ G and < 77^ — 4^ G, 

COS. (/3 + p) ^ COS. (13 — p) ' 

the body will move neither up nor down. 

If a is < p, the force necessary to push the body down the 
plane is 

^ _ sin. (p - a) 



COS. (p + 13) 



G. 



Fig. 259. 



The latter formula can be found by the simple application of 
the parallelogram of forces P Q G, Fig. 259. Since a body 

counteracts any force from another body, 
when the angle of divergence of the di- 
rection of the force from that of the normal 
to the surface is equal to the angle of 
friction p (§ 172), a state of equilibrium 
will exist in the foregoing case, when the 

resultant Q = Q of the forces P and G 
forms an angle N Q — p with the nor- 
mal. If, in the general formula 

P _ sin. G OQ 

G 



sin. P OQ' 
we substitute GOQ= GON^- NOQ = 
ai-p2indiP0Q=P0S + S Q = (3 -{- 
90° — p, we obtain 

sin. (a -\- p) sin. (a -\- p) 




P 

G 



sin. {(3 - p + 90°) COS. {(3 - p)' 



If the force Pi is to prevent the body from sliding down the? 
inclined plane, the resultant Q^ falls on the lower side of the normal 
Nf and the angle of friction p enters in the calculation with a 
negative sign, and consequently we have 

P _ sin. {a — p) 
'G ~ COS. {[3 r pY 



170.] 



RESISTANCE TO FRICTION, ETC. 



3^ 



If the body lies upon a horizontal plane, a is = 0, and the force 
necessary to move it forward becomes 

p _ (f) G _ ^ ^^'^^- P 

~ COS. 13 -{-(f) sin. 13 ~ cos. (/3 — p)' 
If the force acts parallel to the inclined plane, I.E., in the 
direction of its slope, we haye {3 = 0, and therefore 



Ti / . , X ^ sm. (a ± p) 
P = (sm. a ± d) COS. a) G = ^ — 

^ ' COS. p 



G. (Compare § 172.) 



If, finally, the force acts horizontally, we have 

(3 = — a, COS. (3 = COS. a and sin. fi = — sin. a, and consequently 



P = 



sm. a ± (j) cos. a 



G = 



_ tang, a ± </> 
1 =r; tana, a 



COS. a ^ (p sin. a ' " 1 =1= <;f> tang. 
P — tang, (a ± p) G, which is also given by the direct 
resolution of the parallelogram P Q G. 

Farther, the force necessary to push the body up the plane 
becomes a minimum, when the denominator cos. {[3 — p) becomes a 
maximum, that is, when it is — 1, or when (3 — p is = 0, i.e. when 
13 = p. When the angle formed by the direction of the force with' 
that of the inclined plane is equal to the angle of friction, this 
force is a minimum and is P = sin. (a + p) . G. 

Example. — What is the pressure along the axis of a wooden prop 
A E, Fig. 260, which prevents the mass of rock A B C D^ weighing G = 
5000 pounds, from sliding down an inclined plane (the floor of a mine), 
when the inclination of the prop to the horizon is 35°, that of the inclined 
plane G B^ 50° and when the coefficient of friction o is = 0,75 ? Here 
we have 

G = 5000, a = 50°, /? = 35° - 50° = - 15° and </> = 0,75, 
and the formula gives 

sin. a — (p cos^ a ^ sin. 50° — 0,75 cos. 50° 



P = 



COS. jS 
0,766 



^ sin. (3 ' 
0,48Sj 



G = 



0,966 + 0,194 



Fig. 260. 



5000 = 



COS. 15° + 0,75 sin. 15° 
1224 pounds. 



5000 




1,160 

If the prop was horizontal, we would have 
/? = — 50° and tang, p = 0,75, or p = 36° 52', 
from which we obtain 

F = G tang, (a — p) = 5000 tang. (50° — 36° 52') 
:=5000 ta7ig. 13° 8'=5000 . 0,2333 = 1166 pounds. 
In order to push the same mass of rock by 
means of a horizontal force up the floor, whea 
the other circumstances are the same, a force 
P= G tang, (a + p) = 5000 tang. 86° 52' 
= 5000 . 18,2676 = 91338 pounds would 
be necessary. 



326 



GENERAL PRINCIPLES OF MECHANICS. 



[§ I*??. 



177. The normal pressure, with which a body A C presses 
upon the indined plane F H, Fig. 261, while being pushed up it, is 

^r r^ (^ sin. P Q G sin. (90° - a - ^3) 

JS = Q COS. p = —^j- — p ^ ^ COS. p = -Z7ZrT^~:~~aKo IV COS. p 



sin. F Q 
G COS. {a + 13) COS. p 



sin. (i3 + 90° - p) 



COS. {13 — p) 
and, on the contrary, when we prevent its sliding down, we have 

-AT r^ r^ ^ -Ky r^ G COS. (« + /3) COS. p 

N, = Q, COS. Q,ON,^ Q, COS. p - ^((3Tpy 

If the direction of the force is parallel to the direction of the 
plane, we have (3 = and N = G cos. a, and when its direction is 
horizontal, we have (3 = — a and 

G COS. p 



jsr 



COS. {a ± py 



Fig. 361. 




« Fig. 362. 




The normal pressure becomes null, when cos. {a -{- (3) = or 
a + /3 = 90°, and becomes negative, when a + ^3 is > 90° or /3 is 
> 90° — a. In the latter case the inclined plane is not under but 
over the body, as is represented in Fig. 262. Here again the two 
extreoae cases of equilibrium exist when the resultant Q or §i, 
which is transmitted to the inclined plane F H, diverges from the 
normal either above or below it at an angle, which is that of the 
friction JVOQ = WOQ^ = p. 

In the foregoing development of the formulas for the equili- 
hrium of a body upon an inclined plane it is supposed, that the 
resultant Q can be completely transmitted from the body A C to 
the support F H R, which forms the inclined plane ; this is only 



§ 177.] 



RESISTANCE OF FRICTION, ETC. 



327 



Fig. 263. 




possible (according to § 146), when the direction of this force passes 

through the supporting surface 
C D oi the body A C, Other- 
wise the body A G, Fig. 263, has 
a tendency to revolve or overturn 
about the outer edge C, and this 
tendency increases with the dis- 
tance (7 A" = e of this edge from 
the direction § of the result- 
ant Q. 

If a denotes the distance C L 
of the direction P of the force 
and h the distance C E of the 
vertical line of gravity G of 
the body from the outer edge G, 

then the moment, with which the body seeks to turn from left to 

right about G, is Q e = P a — G b. 

P 1) 
If P a were ~ G h oy -^ = -, the resultant Q would pass 

6r CI 

through the edge G and would be counteracted by the inclined 
plane ; if P « were <i G h, the body would have a tendency to turn 
from right to left, which turning would be j)revented by its im- 
penetrability. 

If, on the contrary, P a is > G h the body must receive a second 
support or be guided by a second inclined plane P, H^, If this 
second inclined plane counteracts in A the force N and the fric- 
tion N caused by it, the inclined plane P, H^ will react upon the 
body in A with the opposite forces — N and — </> N, which pre- 
vent the turning of the body about G, and the sum of the moments 
of these forces must be equal to the moment of rotation of the 
force Q, i.e. Nl + (p N d = Q e — P a ~ G b, or 

1) JV (I -{- ct> d) = P a - G b, 

I and d designating the distances G Pand G B of the edge A from G 
in the directions parallel and at right angles to the inclined plane. 

If, further, JV, is the pressure of the body upon the inclined 
plane F H at G and (j) iV, the friction caused by it, we can put 

2) P COS. (3 = G sin. a + (.Y + iV,) and 

3) P sin. I3 — G cos. a -j- X — ^Yi. 
Eliminating iVi from the last two equations we obtain the equa- 
tion of condition. 



828 GENERAL PRINCIPLES OF MECHANICS. [§178. 

P {cos. 13 + (p sin. 13) = G (sin, a -{- (f) cos. a) + 2 iV, 

p (i C h 

and substituting the value N = — —^ from equation (1) we 

have the equation 

P (cos. (3 + (f) sin. (3) = G (sin. a + (p cos. a) + ^ "^ [^^ " ^ ^^ 

I + <f) a 

or P I — ~— (cos. (3 + (j) sin. (3) — cp a\ 

^ /Z -f 6? , . ^ , ^ , \ 

= 6r I (sm. a -i- (p COS. a) — </> bj, 

from which we obtain finally 

p _ (Z + 6?) (sin. a + (p COS. a) — 2 (p b ^ 

~ (I -{- (p d) (cos. i3 -\- (p sin. (3) — 2 (p a 

__ (I -{- (p d) sin. (a 4- p) — 2 (p i cos. p ^ 

~ (I + (p d) COS. ((3 ~ p) — 2 (p a cos. p * 

If JVis = 0, we have P a = G h and 

sin. (a + p) Z> , 

775 { = -, whence 

cos. ((3 — p) a 

p ^ sin, (a + p) ^ 

COS. ((3 — p) ' 
as we found before. 

^ § 178. The Theory of the Equilibrium of Supported 
Bodies referred to the Equilibrium of Free Bodies, — In 

investigating the conditions of equilibrium of a body, taking into 
consideration the friction, we will accomplish more surely our 
object, if we imagine the body entirely free and suppose, that every 
body, with which it comes in contact, acts upon it with two forces, 
viz. : with one force iV, which proceeds from it and is normal to the 
surface of contact, and with another force (p iV, which opposes the 
supposed motion of the point of contact on this surface and which 
is caused by the friction between the two bodies. In this way 
we obtain a rigid system of forces, whose state of equilibrium can 
easily be determined according to the rules given in § 90, as is 
shown in the following special case. 

A prismatical bar A B, Fig. 264, is so placed, that its lower end 
rests upon a horizontal floor C ^and that its upper end leans 
against the vertical wall C V; Sit what inclination B A C = a 
does it lose its equilibrium ? We can here express the reactions 
of the floor upon the body by a vertical force R and by the fric- 
tion <p R, which acts horizontally, and, on the contrary, the reaction 



§ 179.] 



RESISTANCE OF FRICTION, ETC. 



329 



Fig. 264. 



of the wall by a horizontal force JV and by a friction N acting 
upwards. Hence, if G is the weight of the rod acting at its centre 

of gravity S, we have here a system of ver- 
tical forces G, R, ^ N and a system of 
horizontal ones N and ^ R. 

When these forces are in equilibrium, 
we have 

1) G^ R + (t>N, 

2) (f) R = JSTand 




3) G.AE=N.AD^-^N.AC. 
But the arm ^ ^ is = ^ S cos. a = 
h A B COS. a, the arm A D = A B sin. a 
and the arm A C = A B cos. a, hence the 
third equation becomes simply 

^ G COS. a = ]V {sin. a + <^ cos. a). 

Combining the first two equations, we obtain 

G = R + (p' R = (1 + (p') R, whence 
G ' . ^^ G(t> 



R 



and N — 



1 + 0' 1 + 0" 

Substituting this value of JN^in the equation (3), we have 



^ G cos. a = 

1 + 0' 

20 



G 



—^ (sin. a -\- (p cos. a), or 



1+0 
tang. « + 0, 



and the tangent of the required angle of inclination is 

1 + 0^ - 2 0' _ 1-0' _ 1 - tang.^ p 

2 tang, p 



tang, a = 



20 
cos.^ p — sin.'' p 



2 
COS. 2 p 



cotg. 2 p 



2 sin. p cos. p sin. 2 p 
= tang. (90" — 2 p); therefore 
/:BAC=a = dO'-2pandLZABC=l3 = 2p. 

§ 179. Theory of the Wedge. — Friction has also a great 
influence upon the conditions of equilibrium of the wedge (see 
§ 149). Let us suppose, that its cross section forms an isosceles 
triangle A B S, Fig. 265, the acute angle of which A S B = a, 
that the force acts in \h.Q centre M of the back of the wedge A B 



330 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 179. 



Fig. 265 




and at right angles to it and that the body G H K presses with a 
certain force JV against the surface of the wedge B S, while the 

wedge reposes with its 
surface A S upon a 
horizontal plane. The 
body CirX is also in- 
closed in - two guides 
G and K, which com- 
pel it, when the wedge 
is pushed forward upon 
the horizontal plane, 
to rise with the load Q 
in the direction U C 
perpendicular to the surface B S oi the wedge. 

Since the direction of the force P forms equal angles with the 
two surfaces A S and B S oi the wedge, the normal pressures iV, iV, 
and consequently the frictions (f) N, ((> N caused by them, are equal 
to each other, and the forces P, JSf, N, N and iV" must hold 
each other in equilibrium. If we decompose each of the last four 
forces into two components, one parallel and the other perpendicu- 
lar to the direction of the force P, the sum of the forces having the 
same direction as P must, of course, be in equilibrium with P. 
But the directions of the forces N, N form, with the direction M S 

of the force P, an angle 90 — -, and those of the forces (p N,(f) N 
an angle -, and therefore the components of iV, iV^ in the direction 
M S are JY sin. - and N sin. -, and those of iV^and iYare N 

COS. -^f and JV cos. -, and consequently we can put 

P = 2 ]\ sin.- + 20 JSTcos. ^r = 2 JVism. ^ + cos.-). 
Z Z \ Z ZJ 

In consequence of the friction N between the surface B S of 
the wedge and the base of the body G H K, this body is pressed 
with an opposite force — (p JV against the guide G H, which causes 
a friction Pi = 0i . i\^ = 0i iV, which resists the upward move- 
ment of the body G H K\ hence we have 

N- F^= Q ov Nil - 00 = e and 

Q 



N = 



1 - 01* 



§ 179.] 



RESISTANCE OF FRICTION, ETC. 



331 



Substituting this yalue for N in the above equation, we obtain 

the force necessary to raise the weight Q 

20 / . a ^ a\ . X- 1 

I „.^ +0 COS. ^j, approximatively 



P = 



2 Q {1 + </)i) ysin. ^ + <i^ COS. ^j 



= 2 Q ysm. x + </> cos. ^- + 0i stn. -I, 

or putting the coefficient of friction </> along the guides equal to 
that along the surfaces A S and B S of the wedge, we obtain 

2Q / • a ^ A • ^- 1 

P = - — ^ istii. ^ + cos. -I, approximatively 



= 2q[(1 



a 



4 



Fig. 266. 
A M B 




+ 0'') sin. ^ + ^ COS. - 

When a wedge ABC, Fig. 266, is used 
for splitting or compressing bodies, the force 
upon the back A B corresponding to the 
normal pressure Q against the sides A C 
and B Cis 



P = 2 



_ / . a a\ 

Q ystn. o + </> COS. -I. 



Example. — Let the load on the wedge repre- 
sented in Fig. 265 be ^ = 650, the sharpness ot 
the wedge a = 25"^ and the coefficient of friction 
(j) = (p^ = 0,36 ; required the mechanical effect 
necessary to move the load Q ^foot along its guides. 

The force is 

2 650 

{sin. 12^° + 0,36 cos. 12^°) 



P = 



1 — (0,36)^ 
1300 



(0,2164 + 0,36 . 0,9763) 



1 — 0,1296 
1300 737 27 

= o;8704 ^^'^^^^ + ^'^^^^^ =" OWoi ^ ^^^'^ pounds. 
The space described by the load is E B^ = ^i = h ^^ot, and that de- 
scribed by the force is 



BL = s 



„ „ a E E. a 

B B^ COS. - = — — - COS. - 

^ 2 sin. a 2 



2 sm. - 
2 



0,25 

sin. l^° 



= o;2r64 = ^'^^'^ ^""*' 

and consequently the mechanical effect necessary is 

P s = 848,2 . 1,155 = 979,6 foot-pounds. 
If we neglected the friction, the work done would be P s = Q s^ = |- • 
650 = 325 ; consequently the friction nearly triples the mechanical effect 
necessary to raise Q. 



332 



GENERAL PRINCIPLES OF MECHANICS. 



[§180. 



Fig. 267. 



D 



'jy 




E 



§ 180. In the same way we can find the force P required, when 
a wedge ABC, Fig. 267, raises a load Q vertically upwards, while 
moving forward itself upon a horizontal plane HO. Let the 
normal pressure between the wedge ABC and the block D, which 
is pressed vertically downwards by the load Q, be = iV", the normal 
pressure of the wedge upon the support H OhQ — R and the normal 

pressure of the block against the 
guide E EhQ = 8. Then P must bal- 
ance the forces R, (pi R, — N and 
— <p JV, and Q the forces S, (jy^ S, JV" 
and (/) iV. 

If a is the angle of inclination 
A B C of the surface ^ ^ of the 
wedge, we can decompose JV into the 
vertical force W cos. a and the hori- 
zontal force JV sin. a, and JV into 
the vertical force (p N sin. a and the 
horizontal force JSfcos. a, and there- 
fore we can put 

1) P = (f)^R -\r Nsin. a + Ncos. a, 

2) i2 = N cos. a — </) Nsin. a, 

3) Q = Ncos.a—cpJVsin.a—cjy^SaVid 

4) S = JV sin. a ■\- <f) N cos. a. 
From the first two equations we obtain 

P = [{1 — (f> <^i) sin. a -h (({> + (pi) cos. a] iV, 
and from the last two 

Q = [(1 — (f)<i) COS. «—(</) + 02) sin. a] iV; 
and dividing the first by the second, we have 

P _ (1 — 0i) sin. a + (0 4- 0i) cos. a 

Q ~ {1 — (f) (p^) COS. a — {(p + (pi) sin. a 
If = 01 = 02J we have, since (p = tang, p and 

J-— I— = tang. 2 p, 

P _ sin. a + COS. a tang. 2 p _ tang, a + t ang. 2 p 
Q ~ COS. a — sin. a tang. 2 p ~ 1 — tang, a tang. 2 p 

= tang, (a + 2 p). 
If we disregard the friction upon the points of support, we can 
put 01 and 02 — 0, and consequently 

P ^ sm.a + <pcos.a ^ tang.a ±± ^ ,^^y ^^^^^ g ^^^^ 

Q COS. a — (p sin. a 1 — ta7ig. a 



181.] 



RESISTANCE OF FRICTION, ETC. 



3e33 



When the load Q acts at right angles to the surface of the 
wedge, the equations (3) and (4) must be replaced by the following 

Q = JV~ 02 /S'and 

whence Q = (1 — ^s) J^, or inversely, 

Q 



jsr = 



and 



1 — 02 

P _ (1 — 0i) sin. a + (0 + 0,) cos, a 

e ~ 1-002 * 

When is = 01 = 02, it becomes 

— - = sm. a -{- COS. a . tang. 2 p. 

The formula P = Q tang, (a + 2 p) is applicable to the deter- 
mination of the conditions of equilibrium, when two bodies Jlf and JV 

are fastened together by 
^^^- ^^^ means of a key A B, Fig. 

268, I. and II. The force 
P applied to the back of the 
wedge causes the tension, 
with which the two bodies 
are drawn against one an- 
other, 

Q = P cotg. (a + 2 p). 

On the contrary, the 
force, with which we must 
press upon the bottom B 
of the key in order to loosen 
it, I.E. to drive it back in the direction B A, is, since a is neg- 
ative here, 

P^ = Q tang. (2 p — a), 

or substituting the former value of Q, we have 

^ tang. (2 p + a)' 

In order to prevent the wedge from jumping back of itself, a 
must < 2 p. 

§ 181. Coefficients of Friction of Axles. — For axles the 
friction of motion alone is important, and for this reason only the 
results of experiments upon it are given. 




334 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 181. 



TABLE III. 
COEFFICIENTS OF FRICTION OF AXLES, ACCORDING TO MORIN. 



Name of the rubbing bodies. 


Condition of the surfaces and nature of the unguents. 






Anointed and 
moistened with 
water. 


Oil, Tallow, 
or Lard. 




■| d 


r 

<L> 

o 


■£§ 




Bell metal 


upon 


L bell metal . 








0,097 








..^^^ 


a 




cast iron. . . 










0,049 









Wro't iron 




bell metal. 


0,251 


0,189 




0,075 


0,054 


0,090 


0,111 




u 




cast iron. . . 








0,075 


0,054 








Cast iron 




u 

bell metal. . 


0,194 


0,137 
0,161 


0,079 


0,075 
0,075 


0,054 
0,054 


0,065 




0,137 
0,166 


Wro't iron 




lig. vitas . . . 


0,188 






0,125 










Cast iron 




(( 


0,185 






0,100 


0,092 




0,109 


0,140 


Lign'm vitse " 


cast iron . . 









0,116 








0,158 


(( 


u 


lig. vitse. . . 


— 




— 


— 


0,070 


— 







From this table the following practically important conclusions 
can be drawn : for axles, journals or gudgeons of wrought or cast 
iron running in bearings of cast iron or bell-metal (brass), greased 
with oil, tallow or lard, the coefficient of friction ^ 

is, when the lubrication is well sustained, = 0,054, 
and with ordinary lubrication, = 0,070 to 0,080. 

The values found by Coulomb differ in some respects from the 
above. 

Remark. — By his experiments upon mediate friction, by means of the 
friction balance, Him obtained several results, which differ somewhat from 
tbose already known. The axle employed by him, consisting of a hollow 
cast-iron drum 0,28 metres in diameter, and 0,22 metres long, was lubri- 
cated upon the outer surface by dipping it in oil and kept cool by causing 
water to pass through its interior. The bronze bearing (8 of copper and 1 
of tin) was pressed upon it by means of a lever 1| metre long and weigh- 
ing 50 kilogr. while the axle made 50 to 100 revolutions per minute. It 
is easy to see, that in the experiments made with this apparatus the fluidity 
and adhesion of the oil employed as unguent must have played an import- 
ant part, since not only the velocity of revolution, but also the rubbing 
surface was very great compared to the pressure. The velocity at the cir- 



§ 183.] RESISTANCE OF FRICTION, ETC. 335 

cumference of the drum, since its circumference was 72 centimetres and 

since it revolved f to i^ times in a second, was 60 to 120 centimetres, or 24 

to 48 inches, while in machines it is generally but from 2 to 6 inches. The 

horizontal section of the axle was 22 . 23 = 506 square centimetres, and 

consequently the pressure on each square centimetre of this section was 

50 
only ^^ = 0,1 kilogram, i.e. 6,45 . 0,220 — 1,42 pounds upon a square inch, 

while this pressure in ordinary machines is generally more than one hun- 
dred pounds. Hirn's experiments were consequently made under condi- 
tions different from those generally met with in very large and powerful 
machinery, and under which the other experiments, such as, e.g., those of 
Morin, were tried, and therefore the variation in the results obtained is 
perfectly explicable. The principal results of Hirn's experiments are the 
following. 

The mediate friction is dependent not only upon the pressure and the na- 
ture and character of the rubbing surfaces and of unguent, but also upon 
the velocity and upon the temperature of the rubbing surfaces and of the 
surroundings, as well as upon the magnitude of these surfaces. The fric- 
tion is directly proportional to the velocity, when the temperature is con- 
stant ; and if the temperature is disregarded, it increases with the square 
root of the velocity. From other experiments Hirn concludes, that the 
mediate friction is also proportional to the square root of the rubbing sur- 
faces as well as to the square root of the pressure. In regard to the par- 
ticular influence of the temperature, the following formula was given by 
these experiments : 

1,0492«' 

in which t denotes the temperature of the rubbing surface, F^ the friction 
at 0°, and F that at f degrees of temperature. 

One of the principal results of these experiments was the determination 
of the mechanical equivalent of heat. This subject will be treated more at 
length, when we discuss the theory of heat. 

§ 182. Work Done by the Friction of Axles.— If we 

know the pressure E between the axle and its bearing, and if the 
radius r of the axle, Fig. 269, is given, we can easily calculate the 
work done by the friction on the axle during each revolution. The 
friction is F = (f) B, the space described is the circumference 2 n r 
of the axle, and consequently the mechanical effect lost by the 
friction isA = (})E.2nr=2Tr(l)Rr. If the axle makes u 
revolutions per minute, the mechanical effect expended in each 
second is 

L — 2'n <p Rr , -= ^ — ■ = 0,10o . u (p B r. 



336 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 183. 



The work done by the friction increases, therefore, with the 
pressure on the axle, with the radius of the axle and with the 
number of reyolutions. We have therefore the following practical 
rule, not to increase unnecessarily the pressure on the axles in 
rotating machines, to make them as small as possible without en- 
dangering their solidity and durability and not to allow them to 
make too many revolutions in a minute, at least, when the other 
circumstances do not require it. 



Fig. 369. 



Fig. 270. 





By the use of friction-wheels instead of plumber-blocks, the 
work done by the friction is diminished. In Fig. 270 A B is 'd 
shaft, whose journal C E Ei rests upon the circumferences E H 
and El H^ of the wheels (friction-wheels), which revolve around D 
and Di and lie close behind one another. The given pressure R 
upon the shaft gives rise to the pressure 



N= N,= 



R 



2 COS. 



a 
2 



Here a denotes the angle D C Dx included between the lines join- 
ing the centres, which are also lines of pressure. In consequence 
of the rolling friction between the axle C and the circumference of 
the wheels, the latter revolve with this axle, and the frictions N 
and Nx are produced on the bearings D and i)„ the sum of which 

R 



is i^ = (iV + N,) = 



COS. 



a 
2 



If the radius DE=z DxEthe de- 



noted by «! and the radius of the axle by ri, we obtain the force, 
which must be exerted at the circumference of the wheels or at 
that of the axle C to overcome the friction, and it is 



§ 182.] 



RESISTANCE OF FRICTION, ETC. 



337 



F,=- F= -\-^ 



tti 



tti 



COS. 



a 
2 



Fig. 271. 



while, on the contrary, it is = <^ B, when the axle lies directly on a 

step. 

If we neglect the weight of the friction- 
wheels, the work done when these wheels are 

employed is ip = ^- — times as great as 

«! COS. ^ 

/i 

when the shaft revolves in a plumber-hlock. 

If we oppose a single friction-wheel G H, 

Fig. 271, to the pressure R of the axles and 

if we counteract the lateral forces, which in 

other respects can be neglected, by the fixed 

a 




cheeks K and L, a becomes = 0, cos. 

-0 = — . 
«1 



2 



1 and the above ratio 



Example. — A water-wheel weighs 30000 pounds, the radius of its cir- 
cumference a is 16 feet and that of its gudgeon is r = 5 inches ; how much 
force is required at the circumference of the wheel to overcome the friction 
or to maintain the wheel in uniform motion, when running empty, and how 
great is the corresponding expenditure of mechanical effect, when it makes 
5 revolutions per minute ? We can here assume a coefficient of friction 
<p = 0,075, and consequently the friction is <^ i2 = 0,075 . 30000 = 2250 

pounds. Since the radius of the wheel is — '- — = -— = 38,4 times as 

o 

great as that of the gudgeon or the arm of the friction, the friction re- 
duced to the circumference of the wheel is 

_<!> B _ 2250 

~ 38,4 ~ "384 

2 5 TT 

The circumference of the gudgeon is — - ' = 2,618 feet ; and conse- 

quently the space described by the friction in a second is 

2,618 . 5 



= 58,59 pounds. 



60 



= 0,2182 feet, 



and the work done by the friction during one second is 

L = 0,2182 . (t> R= 0,2182 . 2250 = 491 foot-pounds. 
If the gudgeon of this wheel is placed on friction wheels, whose radii 

T 

are but 5 times as great as the radius of the gudgeon, that is, if — ^ = ^, 

the force necessary at the circumference of the wheel to overcome the fric- 
22 



338 GENERAL PRINCIPLES OF MECHANICS. [§ 18S. 

tion would be only ^ . 58,59 = 11,72 pounds and the mechanical effect 
expended but ^^ = 98,3 pounds. But in this case the support would be 
much less safe. , 

§ 183. Friction on a Partially Worn Bearing.— The fric- 
tion of an axle A C B, Fig. 272, upon a bearing, which is partially 
worn, so that the shaft is supported in a single point A, is smaller 
than that of a new axle, which touches all points of its bearing. 

If no rotation takes place, the axle presses 
Fig. 273. upon the point B, through which the direction 

of the resulting pressure R passes ; if the shaft 
begins to rotate in the direction A B, the axle 
rises in consequence of the friction on its 
bearing, until the force S tending to move it 
down balances the friction F. The result- 
ant R is decomposed into a normal force N 
and a tangential one S, N is transmitted to 
the plumber block and produces the friction F = (^ N, which acts 
tangentially, 8, however, puts itself in equilibrium with F, and 
we have, therefore, 8 — (p N. According to the theorem of Pytha- 
goras, we have R^ = N'- + 8''-, whence 

■ i?^ = (1 + 00 N^ 
or inversely the normal pressure 

N = — — and the friction F ~ 




or introducing the angle of friction p or cp = tang, p 

„ R tang, p „ , „ . 

F — — -— -- R tang, g cos. p = R sm. p. 

y 1 + tang J p 

Consequently, when the shaft begins to turn, the point of pres- 
sure B moves in its bearing in the opposite direction through an 
angle A C B = the angle of friction p. 

The moment F . C A = F r of the friction on the axle is 
naturally equal to the moment R r sin. p of the pressure R upon 
the bed, both being referred to the axis of revolution C. If there 
were no motion, we would have 

F = (p R = R tang, p = ; 

the friction after the motion begins is cos. p times as great as 
before. Generally = tang, p is scarcely y^o ^^^ ^^^- P > 0,995, 
60 that the difference is scarcely -^-^^q — ~^\-q ; we can, therefore, 
in ordinary cases neglect the effect of the motion. 



§ 184.] 



RESISTANCE TO FKICTION, ETC. 



339 



Fig, 







Fig. 274. 



% If the wheel A B revolves with a nave, Fig. 
273, about a fixed axle A C, the friction is the 
same as if the axle moved in an ordinary plumber- 
block, but when the nave is worn the arm of the 
friction is not the radius of the shaft, but that of 
the opening in the nave. 



§ 184. Friction on a Triangular Bearing. — If we lay the 

axle in a prismatical bearing, we have more pressure on the bearing, 
and consequently more friction than, when the bearing is circular. 

If the bearing A D B, Fig. 274, is tri- 
angular, the axle is supported at two 
points A and B and at both of them 
friction must be overcome. The result- 
ing pressure R is decomposed into two 
components Q and ft, each of which is 
again denomposed into a normal stress 
iYor iV"i and into a tangential one, which 
equal to the friction F = (p ]V and 
Fi = (p aVi. According to the foregoing paragraph, we can put 
these frictions = Q sin. p and Q^ sin. p, consequently the total 
friction is F + F, = (Q + ft) sm. p. 

The forces Q and Qi are found, by the resolution of a parallelo- 
gram of forces formed of Q and Q^, with the aid of the resultant E, 
of the angle of friction p and of the angle A C B = 2 a, corespond- 
ing to the arc A B included between the two points of contact; 
now we have 

QOB = ACD-CAO=:a-psiud 
Q, R = B CD -{- OB == a -i- pand therefore 
QOQi = a — p-{-a + p = 2a. 
By employing the formula of § 78, we obtain 
8171. (a 




c.= 



Pl , ji ^^A Q = "^^P^K B ; 



sin. 2 a sin. 2 a 

ffhence the required friction is 

R '^in o 
F -^ F^ = {Q -{- Q,) sin. p = {sin. [a — p] + sin. [a + p]) -. — —-. 

sm, /i CL 

But from trigonometry we know, that^w. (a — p) + sin. (a + p) 

= 2 sin. a cos. p, and tliat sin. 2 a = 2 sin. a cos. a, and we can 

therefore put 

2 sin. a R sin. p cos. p R sin. 2p 



F + F. 



2 sin. a cos. a 



2 cos. a 



340 GENERAL PRINCIPLES OF MECHANICS. [§185. 

which, owinff to the smallness of o, we can make =^ —. When 

COS. a 

a triangular bearing is used, the friction becomes times greater 

COS» Q/ 

than when a circular one is employed. If, e.g., A D B is 60°, 
A C B is 180" -■ eO** = 120" and ^ G D = a=z 60", we have 

wTTn times = twice as much friction as for a circular bearinsr. 

COS. 60^ ^ 

§ 185. Friction of a New Bearing. — By the aid of the latter 
formula we can find the friction on a new circular bearing, when 
the axle is supported at all points. Let A D B, Fig. 275, be such 

a bearing. Let us divide the arc ABB along 
^^^^^^' which the bearing and axle are in contact into 

very many parts, such as ^ iV, iV^ 0, etc., whose 
projections upon the chord A B are equal, and 
let us suppose that each one of these parts 
transmits from the axle to the bearing equal 

7? 

portions — of the whole pressure E, Here n 

denotes the number of these parts. According 
to the foregoing paragraph, the friction of two 
parts JV" and iVj Oi opposite to each other is 

_ R sin. 2 p 

"" n ' COS. N C U 

N P 
But COS. N C D is also = cos. N P — ^^^j N P represent- 
ing the projection of the part i\^ on ^ ^, and therefore 

,^ „ chord A B 
N P = — —- • 

n 

consequently the friction corresponding to these two parts N and 

•^1 0, is _ Esin.2p n.WO _ Rsin.2p j^ 

~ n ' chord ~ chord 

In order to find the friction for the entire arc A I) B, we have 

only to substitute instead of iV^ the arc AD— ^ADB, for the 

R siTii. 2 o 
sum of all the frictions is equal to — ^ — ^ — . the sum of all the 

^ chord 

parts of the arc ; consequently the friction on a new bearing is 

D • o arc ^ i> 
F= R sm. 2 . -, — T . -r, , 
^ chord A B 

or putting the angle at the centre A B corresponding to the aro- 

contained in the bearing = 2 a^ and the chord A B = 2 A C sin. a^ 

we have 




§186.] RESISTANCE OF FRICTION, ETC. S41 

„ E sin.2p a • i- 1 

F = 5 — -- . —. — or approximatively, 

assuming 2 p = 2 sin. p, 

a 

F = R sin. p . 



sm. a 

Hence the initial friction increases with the depth, that the axle 
is sunk in its bearing, e.g., if the bearing includes the semi-circum- 
ference of the axle, we have a == ^ tt and sin. a — 1, and therefore 

F = - . E sin. p is ^ = 1,57 times as great as it is when a bearing 

2 /y 

has been worn. If the axle does not lie deep in its bearing, or if a is 
small, we can put sin. a — a — ~ = ayl — —\, whence it fol- 
lows that F — \1 -\- p'l ^ '^^'^' P or = i? sin. p, when a is very 
small. 

(§ 186.) Poncelet's Theorem. — The pressure R on the bear- 
ings is generally given as the resultant of two forces P and Qy 
which act at right angles to each other, and it is consequently 

— V P"^ + §\ So far as we need it for the determination of the 
friction 

F=(P R^ cp V~P^T'Q\ 

we can content ourselves with an approximate value of V P^ + Q'y 
partly because an exact value of the coefficient can never be 
given, as it depends upon so many accidental circumstances, 
partly, also, because the product i? is generally but a small 
fraction of the other forces, which act on the machine, e.g., the 
lever, pulley, wheel and axle, etc., which is supported by the bear- 
ings. The formula for calculating the approximate value of 

V P'' + Q" is known as Poncelet's theorem, and its truth can be 
demonstrated in the following manner. We have 

in which x = ^, and if Q is the smaller force, ^ is a simple frac- 
tion. Now let us put V^-\- x^ = fi -^ V X, and let us determine 
the coefficients /x and v corresponding to certain conditions. The 
relative error is 

Vl+x^ — ii~vx ^ u + V X 

y ^ :,-== = 1 - • 



Vl -^ x' V\ + 



X 



342 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 186. 



Fig. 276. 



o 




p 


\ 


\K r ^^ 


A 




^^-^^^ B 



This equation corresponds to the curve S F, Fig. 276, whose 
ordinate, when the abscissa a;=:0, is^ —y — i — ^^ and, when 

the abscissa^ ^ = 1, is «/ = 1 - ^-^^. The curve also cuts the 

axis of abscissas in two points K and N and at 8 lies at its greatest 

distance C S from this axis. If 
we put y --- or 

Vl + x' = II ■\- V X, 

and solve the equation in relation 
to X, we obtain 

Q 

_ ju 1^ =F \^fJ^' + v^ — i 

1 — V 

the values of which are the abscissas A K and ^ iV^ of the points 
^and N, where the curve cuts the axis, and also those values for 
which the error is = 0. In order to find the abscissa ^ C of the 
maximum negative error C 8, we must put the differential ratio 
dy _(ii ^ V x) {1 -\- rr')-? X — V {1 + Ofi'Y^ 
dx 1 + x'' 

(see Article 13 of the Introduction to the Calculus). 
This condition is fulfilled by putting 

ifi + V o:) (1 + x-)—\x — V {1 -{- x^Y- or 

V 

{fl + V x) X ~ V {i + x")^ I.E. x = -. 
' fl 

V 

According to this formula, the abscissa A C — - gives the greatest 

^^ 

negative ordinate. 



=: 



U + V . 



V 



w I f^' + v^ \ 



1). 



V I + 



11' 



In order to have neither a great positive nor a great negative 
error, let us put the three ordinates A = 1— fj^, BF = 1 — 

U ^ V 



V2 



and C 8 = Vii" + i^^ — 1 equal to each other, and deter- 



mine from them the coefficients fi and v. We have 

H — ^—=z.-, I.E., V — {V'^ — l)}-i — 0,414 iJb and 



2 - fji = Vij,' + v% LE., 2 = ^(1+ Vl -{- 0,414=^) 
and consequently 



§ 187.] RESISTANCE OF FRICTION, ETC. 343 

^ :^ ? =: 0,96 and v = 0,414 . 0,96 = 0,40. 

1 + V 1,1714 



We can, therefore, put V 1 -\- x" = 0,26 + 0,40 . x, and in like 
manner the resultant 

B = 0,96 P + 0,40 Q, 
iind we know that in this case the greatest error we can make is 
±y = l— jj, — 1 — 0,96 = 0,04 = four per cent, of the true 
yalue. 

This formula supposes, that we know, Avhich of the tyro forces 
is the greater ; if this is unknown to us, we assume 

VTT'x' = /i (1 + x) 

and obtain in that way 

_^(1J-^^ 

In this case not only x = 0, but also ic = oo gives an error 
1 — |U. If we put X = - — 1, we have the greatest negative error 

Putting these errors equal to each other, we obtain 

1 _ , ^ , ^Y _ 1, or , = --^^. = ^^ = ^^^ = 0,838. 

In case we do not know, which of the forces is the greatest, we 

can write 

E = 0,83 (P + Q), 

then the greatest error we can make is ± ^ = 1 — 0,83 = 17 per 

cent. = I of the true value. 

If, finally, we know that x is not over 0,2, we do best to neglect 

X altogether and to put V F"" + g"* = P ; if, however, x is over 
0,2, it is better to make 



V P' -h Q' = 0,888 P -\- 0,490 . Q. 
In both cases the maximum error is about 2 per cent.* 

§ 187. The Lever. — The theory of friction just given is appli- 
cable to the material leA'cr, to the wheel and axle and to other 
machines. Let us now take up the subject of the lever, discussing 
at once the most general case, that of the bent lever A C B, 

* Polytechnisclie Mittheilungen, Vol. I. 




344 GENERAL PRINCIPLES OF MECHANICS. [§ 187. 

Fig. 277- Let us denote, as formerly (§ 136), the arm of the lever 

C^ of the power P by a, the 
lever arm C B of the load Q\>yb 
and the radius of axle by r, and 
let us put the weight of the lever 
= G, the arm C E of the same 
=: s and the angles A P K and 
B Q K formed by the directions 
of the forces with the horizon 
= a and j3. The power P produces the vertical pressure P sin. a 
and the load Q the vertical pressure Q sin. (3, and the total vertical 
pressure is F = 6^ + P sin. a + Q sin. (3. The force P produces 
also the horizontal pressure P cos. a and the load an opposite 
pressure Q cos. (3, and the resulting horizontal pressure is H =^ 
P COS. a — Q COS. (3, and the total pressure on the axle is 
E = fi V + vH = iJi{G i-Psin. a +Q sin. (3) -f v (P cos. a — Q cos. 13) 
in which, however, the second part v (P cos. a — Q cos. 13) is never 
to be taken as negative, and, therefore, when Q cos. /3 is > P cos. a 
the sign must be changed, or rather P cos. a must be subtracted 
from Q COS. 3. In order to find the value of the force correspond- 
ing to a state of unstable equilibrium so that for the smallest addi- 
tion of force motion will take place, we put the statical moment 
of the power equal to the statical moment of the load plus or minus 
the moment of the weight of the machine (§ 136) and plus the 
moment of the friction ; thus we have 
Pa= Qb±: Gs + (f) Er 

=zQJ)±Gs + (l){fjiV+v H) r, whence 
P - Qb ±G s + <p[fi{G + Q sin. (3) ^^ v Q cos. i3] r 
a — II ^ r sin. a ^ v (p r cos. a 
If P and Q act vertically, we have simply E — P + Q + G 
and therefore Pa=Qh±Gs + (t){P+Q + G)r. If tLe lever 
is one armed, P and Q act in opposite directions to each other and 
E is ^P — Q + G and therefore the friction is less. But E 
must always enter into the calculation with a positive sign, for the 
friction E only resists motion and never produces it. We see 
from this, that a single armed lever is mechanically more perfect 
than a double armed one. 

Example. — If the arms of the bent lever represented in Fig. 377 are 
« = 6 feet, 5 = 4 feet, s = ^ foot and r = 1| inches, if the angles of incli- 
nation are a = 70°, (S = 50^ and if the load is Q = 5600 pounds and the 
weight of the lever G \s = 900 pounds, the force necessary to produce 



§188.] RESI-5TANCE OF FRIC'i;iON, ETC. 345 

unstable equilibrium is determined as follows. The friction being disre- 
garded, we have P«+ G s = Qb and therefore 

„ Qh-Gs 5600.4-900.^ ^^^^ 

P = = ~ = 3658 pounds. 

If we put /I = 0,96 and v = 0,40, we obtain 

fi{a + Qdn. 0) = 0,96 (900 + 5600 dn. 50") = 4983 pounds, 
V Qcos. fi = 0,40 . 5600 cos. 50° = 1440 pounds, 
// sin. a = 0,96 . sin. 70" = 0,902 and 
vcos.a=^ 0,40 . COS. 70" = 0,137. 
It is easy to see, that P cos. a is here smaller than Q cos. ,5 ; for since P is 
approximatively 3658 pounds, we have P cos. a = 1251 pounds, while, on 
the contrary, Q cos. /3 is = 3600 pounds ; therefore we must employ in this 
case for v Q cos. p and forvcpr cos. a the lower sign and put 
_ 5600 . 4 — 900 . I + (j) r (4982 + 1440) 
~ 6~- (/) r (0,90"2 - 0,137") * 

Assuming the coefficient of friction (p = 0,075, we obtain 

r = 0,075 . ^ = 0,009375 and 6422 (/> r = 60 
and the force required 

„ 22400 - 450 + 60 22010 „^^„ 
P = 6 - 0,0 717— = ^928 = ^^^^ P^^^^^' 
Here the vertical pressure, when we substitute the force P = 3658 pounds 
determined without reference to the friction, is 

V = 3658 sin. 70" + 5600 sin. 50^ + 900 = 3437 + 4290 + 900 
= 8627 pounds, 
and, on the contrary, the horizontal pressure is 

H= 5600 cos. 50 — 3658 cos. 70 = 3600 — 1251 = 2349 pounds. 
Here 5"is > 0,2 V, and therefore we have more correctly 

B = 0,888 . JS'+ 0,490 V= 0,888 . 8627 + 0,490 . 2349 = 8811, and 
consequently the moment of the friction is 

= (f>rP= 0,009375 . 8811 = 82,6 foot-pounds ; 
and finally the force 

22400 - 450 + 82,6 ^^^^ 
P = ^ = 3672 pounds, 

which value differs very little, it is true, from the one obtained above. 

§ 188. Friction of a Pivot. — If in a wheel and axle there is 
a pressure in the direction of the axis, which is always the case, 
when the axle is vertical, in consequence of the weight of the 
machine, friction is produced upon the base of one of the journals. 
Since there is pressure at all points of the base between the pivot 
and the step (or footstep), this friction approaches nearer to the 
ordinary friction of sliding, than to what we have previously con- 
sidered as axle friction, and we must therefore employ in this case 
the coefficients of friction given in Table II. (page 320). In order 



346 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 188. 



Fig. 378. 



to find the work done by this friction, we must know the mean 
space described by the base A B, Fig. 278, of such a pivot. We 
assume that the pressure E is equally distributed over the whole 
surface, that is, we suppose that the friction upon equal portions 
of the base is equally great. If we divide the base by means of the 
radii C D, B, etc., in equal sections or triangles, such as D C E, 
these correspond not only to equal frictions, but 
also to equal moments, and we need therefore only 
find the moment of the friction of one of these 
triangles. The frictions on such a triangle can be 
considered as parallel forces, since they all act 
tangentially, I.E., at right angles to the radius C D ; 

(I >w and since the centre of gravity of a body or of a 
R ! surface is nothing else than the point of application 
^^^^ ) of the resultant of the parallel forces, which are 
equally distributed over the body or surface, we can 
consider the centre of gravity S of this sector or 
triangle D C E as the point of application of the 
resultant of all the frictions upon it. If the pressure on this sector 

D 

is = — and radius CD 




C E = r,ii follows (according to § 113), 
that the statical moment of the friction of this sector is 



n ^ n 



M = n , % r -- 



= ^ (p R r. 



and finally that the statical moment of entire friction of the pivot is 

(})E 
n 
Sometimes the rubbing surface is a ring ABED, Fig. 279, 
If the radii of the same are C A = r^ and C D — 
ro, we have here to determine the centre of gravity 
/S^ of a portion of a ring. Hence, according to 
8 114, the arm is 



Fig. 279. 




C 8 



2 

3 ^. 



r 



and therefore the moment of the friction is 



M=l R 



r{ — ^2 



If we introduce the mean radius — -x — = 7' 

and the breadth of the ring ri — rg = l, we obtain also for the 
moment of the friction 



189.] 



RESISTANCE OF FRICTION, ETC. 



347 



M= (p E 



{' + iIt)- 



The mechanical effect of the friction is, in the first case, 
A = 2n.f(pEr = {TT(pRr, and, in the second case. 

From the above data it is easy to calculate the friction upon st 
journal composed of one or more collars, when a vertical shaft is 
borne by it. It is also easy to see, that, in order to diminish the 
loss of mechanical effect, the pivots should be made as small as 
possible, and that, when the other circumstances are the same, the 
friction is greater on a ring than on a full circle. 

Example. — A turbine, weighing 1800 pounds, makes 100 revolutions 
per minute, and the diameter of the base of the pivot is 1 inch ; how much 
mechanical effect is consumed in a second by the friction of this pivot ? 
Assuming the coefficient of friction = 0,100, we obtain 

9 i? = 0,100 . 1800 = 180 pounds, 
the space described in a revolution is 

= f 77 r = I- . 3,14 . ^V = 0,1745 feet, 
and therefore the work done in one revolution is 

= 180 . 0,1745 = 31,41 foot-pounds. 

But this machine makes in a second i^^ = |- revolutions, and therefore 

the required loss of mechanical effect is 

B14,l 
= — ^ — = 52,3 foot-pounds. 

§ 189. Friction on Conical Pivots, — If the end of the axle 
A B D, Fig. 280, is conical, the friction is greater than when the 

pivot is flat, for the axial pressure R is 
decomposed into the normal forces N, Ni, 
etc., which produce friction and whose 
sum is greater than R alone. If half the 
angle of convergence ADC— BDC=a^ 
we have 

sm. a 
and therefore the friction of this conical 
pivot is 

F= (f) 



Fig. 280. 




R 



sm. a 



If we denote the radius C A = C B of the axle at the place of 
entrance in the step by r„ we have, in accordance with what pre- 
cedes, the statical moment. 



348 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 189. 



M 



(pR 



^1 = 10 



sin. a 



or, since 



1 



sm. a 



sm.a 

C A 

= the side DA of the cone 



sm. a 



a, we have 



M^l^ Ra. 



If we allow the axle to penetrate a very short distance into the 
step, the friction is less than for a flat pivot, and for this reason 
we can employ conical pivots with advantage. If, e.g.. 



7*1 r 

a — —. — - = -, or ri = :V 7* sm. a, 
sm. a 2 "^ 



1 



the conical pivot, whose radius is ri, occasions only half as much 
loss of mechanical effect as the flat pivot, whose radius is r. 

If the pivot forms a truncated cone. Fig. 281, friction is pro- 
duced on the conical surface and on the flat base, and we have for 
the statical moment of the friction 



\ sm. a / 



R 



r 



2 J 



when r denotes the radius C A at the point, where the pivot enters 
the step, Ti the radius of the base and a° half the angle of conver- 
gence. In consequence of the great lateral pressure JV the step 
becomes soon so worn that finally only the pressure on the base 
^ F remains and the moment of the friction becomes M — j (p R r^. 

Fig. 281. Fig. 282. Fig. 283. 






Vertical shafts or pivots are very often rounded off as in Figs. 
282 and 283. Although by this roundiug the friction is not in 
any way diminished, yet a diminution of the moment of the fric- 
tion can be produced by diminishing the penetration of the pivot 
into the step. If we suppose the rounded surface to be spherical, 
we obtain with the aid of the calculus, for a hemispherical step the 
moment of friction 

Jf=^-. Rr; 

and for a step forming a low segment approximatively 

Jf = I [l -f- 0,3 (^yj R r„ 



190.J 



RESISTANCE OF FRICTION, ETC. 



349 



in which formula r denotes the radius of the sphere M A = M B 
and ri the radius of the step C A = C B. 

Kemabe. — The pressure R upon the centre ABB, Fig. 384, of the 

spindle of a turning-lathe is perpendicular to 
Fig, 284. the direction of the axis I) X and is decom- 

posed into a normal pressure JSF and a lateral 
pressure /S parallel to the axis. Retaining the 
same notation, that we employed above for 
conical pivots, we have 

iV = and /S = i2 tana a. 

COS. a 




The moment of the friction caused by if is 



COS. a 



or since r^ = C A = D A sin. AD G = a sin. a, when a denotes the length 
C D oi the portion of the centre which is buried, we have M = ^ (j) Ba 
tang. a. 

The lateral force S is entirely or partly counteracted by an opposite 
force /S'j on the other centre. 

Example. — If the weight of the shaft and other parts of a whim gin is 
H = 6000 pounds, the radius of its conical pivot is = r = 1 inch and the 



Migle of convergence 2 a of the latter is 

of the friction is 

,, o Br „ ^, 6000 1 



90", the statical moment 



100 



^Vi 



47,1 foot-pounds. 



stn. a « ' sin. 45" ' 12 
If the shaft in hoisting a bucket out of a mine makes u = 24: revolu- 
tions, the mechanical effect consumed by the friction of the pivot during 
this time is 

^ = 3 TT w . I ^ -T^ = 2 TT . 24 . 47,1 = 7103 foot-pounds. 

§ 190. The so-called Anti -friction Pivots. — Supposing 
that the axial pressure on a pivot ABBA, Fig. 285, is propor- 
tional to the surface of the cross- 
section, we can put the vertical 

pressure per square inch Bx = -^, 

B being the total pressure and G 
the area of the vertical projection 
ADDA of the whole rubbing 
surface ABBA, If now a is the 
angle of inclination C T of the 
element of the surface to the 
axis T oi the pivot, the normal 
pressure on each square inch 







350 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 190. 



R. 



sm. a 



and the corresponding friction 



(b 



R, 



R 



of the bearing, will be JVi = 
will be 

sm. a (jr sm. a 

and if y denotes the distance or radius of friction M 0, the moment 
of this friction is 

T7 A R 



y 



G 



or, smce 



y _ 



sm. a 



tangent T, 
F,y=(p 



G 



sm. a 



T. 



Fig. 386. 



In order to obtain a regular wearing away of the axle and of its 
step, the moment F^ y must be the same for all positions, and con- 
sequently the tangent T must have the same value for all points 
of the generating curve A B oi the axle, and therefore the mo- 
ment of the friction on the whole pivot is when T = a 

M= F,y . G = (p Ra. 

The curve A B, whose tangent T, measured from the 
point of tangency to the axis O X, is constant, is a tractrix or trac- 
tor y, and is generated by drawing a heavy point A, Fig. 286, over a 

horizontal plane by means of a string, 
whose end moves along a straight line 
C X. This string forms the constant 
tangent lines A = al=l32 = y3, 
etc. = a. In . order to construct this 
curve, we draw C A = a perpendicular 
to the axis C X and take in C A, a 
near to A, and lay off a 1 r= ^^^ take /? 
in a 1, near to a and lay off (3 2 = a, 
here again take y near to /3 and lay off 
y S = a, etc., and we then draw a curve 
tangent to the sides Aa,a [3, (3y,y6 . . ., 
etc. This method gives the tractory 
the more accurately the smaller the 
sides A a, a [3, [3 y, y 6 . . ., etc., are. 
Schiele calls this curve the anti=friction 
curve. (See the Practical Mechanics' 
Journal, June number, 1849, translated in the Polytechnisches 
Centralblatt, Jahrgang, 1849.) 

If, as is represented in Fig. 285, we make the anti-friction curve 




§ 190.] RESISTANCE OF FRICTION, ETC. 351 

end at the circumference of the shaft the maximum radius of 
friction C A = r is at the same time the constant tangent a, and 
therefore the moment of the friction M — (f> E r is independent of 
the length of the pivot. When the rubbing surface is flat and of 
the same radius, the moment of friction is i¥i == | i? r, that is, 
one third smaller, and it decreases still more in time ; for the exte- 
rior portions are more worn than the interior ones, and thus the 
surface of friction becomes less. 

The plugs mid chambers of cocJcs are sometimes made in the 
form of the anti-friction curve ; for in this case the conditions are 
the same as in a pivot. 

Remark. — When the pressure E on the pivot is so distributed that the 
amount of the wearing, measured in the direction of the pressure, is equal 
in all points of the circumference of the pivot, we have 

sin.a^- sin.a^ sin. a^ 
and for conical pivots, where 

Oi =: ag = ag . . . = a ; N^y^=N^y^=N^y^,.. 
If Oj, O3, O3 . . . denote the surfaces, upon which the normal pressures 
.2^1, iV^g, iVg . . . act, we have 

R = N^ Oj dn. ttj + iVg O3 8in. a^ + iVg O3 sin. a^ -{■ . . . 
or for conical pivots i? = (JV^ O^ + iVg O3 + iVg O3 + . . .) sin. a. 

The portions of the surface can be considered as rings of the same 

height -, whose widths are — -. , and whose radii are 2^i, ^g? 2^3) conse- 
quently we have 

0. = 3 7r y^ — -. , O3 = 2 TT 2/0 -—. , O3 = 2 7T 2/0 —-. , etc. 

^ ^^ n sin. a' "* ^^ n stn. a' ^ ^^ n sm. a' 

0^ = — Oj, O3 = — Oj, etc., and also 
Vx Vi 

N^ Oi = ifg Og = i^3 ^3 • • •, and i? = 71 . N^ 0^ sin. a. 

Therefore, under the above assumption, the normal pressure on the 

equally high rings of the circumference of the pivot are equal. 

r> 

Inversely we have N^ 0^ = — -. — , hence the moment of the friction 

n sin, a 

on the pivot is 
M=^{N^ 0, y^ + N^ Og yg + i^3 O3 2^3 + . . 

= -^ ^1 ^1 (2/1 + yg + . . . + 2/n) = ^^-^^^^ (2/1 + 2/2 + ... + yn). 

If we have a truncated conical pivot, whose radii are r^ and rg, we must 

n (v + 7* ") 
put 2/1 + 2/2 + . . . + yn = ^ ^, from which it follows that M = 

</> jg (r^ + rg) 
2 sin. a 
For a complete conical pivot, whose radius is r^ — 0, we have M = 



352 GENERAL PRINCIPLES OF MECHANICS. [§191. 

<^ R r^ H y 

5—1 — , while in a foregoing paragraph (§ 189) we found Jf= f ^ — ; — >-. 
i Sin. a sin. a 

See the article by Mr. Reye upon the Theory of Friction of Axles in Vol. 

6 of the Civihngenieur, as well as the article upon the same subject by 

Director Grashof in the 5th volume of the Journal of the Association of 

German Ingenieurs. 

§ 191. Friction on Points and Knife-Edges. — In order to 
diminish as much as possible the friction of the axles of rotating 
bodies, they arc often supported on sharp points, knife-edges, etc. 
If the bodies employed were perfectly solid and inelastic, no loss of 
mechanical effect in consequence of the friction would take place 
by this method, since the space described by the friction is immeas- 
urably small ; but since every body possesses a certain degree of 
elasticity, upon placing it upon the point or knife-edge, a slight 
penetration takes place and a surface of friction is produced, upon 
which the friction describes a certain space, which, although small, 
occasions a loss of mechanical effect. When the rotation or vibra- 
tion of a body supported in this way has continued some time, such 
surfaces of friction are arcs developed by the wearing away of the 
point or knife-edge, and the friction is then to be treated as we have 
previously done. This mode of support is therefore only employed 
in instruments such as compasses, balances, etc., where it is impor- 
tant to diminish the friction and where the motion is not constant. 

Coulomb made experiments upon the friction of a body sup- 
ported by a hard steel point and movable around it. According 
to these experiments, the friction increases somewhat faster than 
the pressure, and changes with the degree of sharpness of the 
supporting point. It is a minimum for a surface of garnet, greater 
for a surface of agate, greater for a surface of rock crystal, still 
greater for a surface of glass, and the greatest for a steel surface. 
For very small pressures, as, e.g., in the magnetic needle, the point 
can be sharpened to an angle of convergence of 10° to 20°. If, 
however, the pressure is great, we must employ a much larger 
angle of convergence (30° to 45°). The friction is less, when a body 
lies with a plane surface upon a point than when the point plays in 
a conical or spherical hollow. The circumstances are the same for 
a knife-edge such as that of a balance. Balances, which are to be 
heavily loaded, have knife-edges with an angle of convergence of 
90°. When the balance is light, an angle of 30° is sufficient. 

If we assume that the needle A B, Fig. 287, has pressed down 
the point F C G an amount D G E, the height of which G M—li, 
and the radius of which D M — r, and if we suppose the volume 



§ 192.] 



RESISTANCE OF FRICTION, ETC. 



353 



I TT r^ h tohe proportional to the pressure R, the measure of the fric- 
tion can be found in the following manner. If we put I n r"" h = 
fi R, in which jit is a coefficient given by experiment, and substi- 
tute the angle of convergence DCE = 2aoTh = r cotg. a, we 
obtain for the radius of the base 



From this we see that we can assume, that the friction on a pivot 
increases with the cube root of the fourth power of the pressure 
and with the cube root of the tangent of half angle of convergence. 

Fig. 287. Fm. 288. 





The amount of friction of a beam A B, Fig. 288, oscillating on 
a knife-edge C d, can be found in like manner. If a is the half 
angle of convergence D C M,l the length C Cy of the edge and R 
the pressure, we have 



(t> Rr 



= ./(^ 



tang, af 



§ 192. Friction of Rolling. — The theory of rolling friction 
is as yet by no means established upon a firm basis. We know, 
that the friction increases with the pressure, and that it is greater, 
when the radius of the roller is small than when it is large ; but 
we cannot yet give the exact algebraical relation of the friction to 
the pressure and to the radius of the rolling body. Coulomb made 

a few experiments with rollers of 
lignum-vitse and elm from 2 to 10 
inches thick, which were rolled 
upon supports of oak by winding 
a thin string around the roller and 
attaching to the ends of it the un- 
equal weights P and Q, Fig. 289. 
According to the results of these 
experiments, the rolling friction is 
directly proportional to the pressure 
and inverselv to the radius of the 




354 



GENERAL PRINCIPLES OP MECHANICS. 



[§192. 



rollers, so that the force necessary to overcome the rolling friction 

7? 

can be expressed by the formula F = f . — , E denoting the press- 
ure, r the radius of the roller and / the coefficient of friction to be 
determined by experiment. If r is given in English inches, we 
have, according to these experiments, 

I'or rollers of lignum-vitae, / = 0,0189 
For rollers of elm, / = 0,0320. 

The author found for cast-iron wheels 20 inches in diameter, 
rolling on cast-iron rails, 

/ = 0,0183, and Sectionsrath Eittinger 
/= 0,0193. 
According to Pambour, we have for iron railroad wheels about 
39,4 inches in diameter 

/ = 0,0196 to 0,0216. 

7? 

The formula F = f — supposes that the force F, which over- 
comes the friction, acts with a lever-arm H C = IT L = r equal to 
'the radius of the roller, and that it describes the same space as the 
latter. If, however, it acts on a lever arm^^ = 2 r, the space 
described by it is double that described by the roller on the sup- 
port, and the friction is therefore 

F.-=^lF=f^. 

^ 2 r 

The conditions of equilibrium of rolling friction can be found 
in the following manner. In consequence of the pressure Q of the 
roller A C B upon the base A 0, Fig. 290, the latter is compressed ; 
the roller rests, therefore, not upon its lowest point A, but upon 
the point which lies a little in front of it. Transferring the 
points of application A and B of the forces Q and F, of which the 
latter F is the force necessary to overcome the friction, to their 

point of intersection D, and constructing 
with Q and F the parallelogram of forces, 

we obtain in its diagonal D R the force 
R, with which the roller presses upon its 
support in 0, and' it Is therefore necessary 
that the moments of the forces of the bent 
lever A i\^ shall be equal to each other. 
If we put the distance N of the point 
of support from the direction of the 
force = a, and the distance M of the 
same point from the vertical line of grav- 



FiG. 290. 
D 




§ 192.J 



RESISTANCE OF FRICTION, ETC. 



355 



ity of the body = /, we have 

Fa= Qf, 
from which we obtain the required equation 

_/ 



F 



a 



Q- 



The arm /is a quantity to be determined by experiment and is 

so small, that we can substitute instead of a the distance of the 

lowest point A from the direction of the force F, as well as instead 

of § the total pressure i?. 

f 
Hence we have F ^^ "^ R, and consequently, when the force 

acts horizontally tind through the centre C, <z =: r or 

/ 



F 



r 



R, 



and on the contrary, when this force acts tangentially at the high- 
est point ^ of the roller, ^i _ f 



F = 



2r 



R. 



Fig. 201, 



J— ^P 



The so-called coefficient of friction / of rolling friction is there- 
fore no nameless ([uantity, but a line, and must therefore be ex- 
pressed in the same unit of measure as a. 

If a body A S B is placed upon two rollers C and D, Fig. 291, 
and moved forward, the force P required to move the body is very 

small, as we have only two rolling 
frictions to overcome, viz., one 
between A B and the rollers and 
the other between the rollers and 
the surface H K. The space de- 
scribed progressively by the roll- 
ers is but one-half that described 
by the load R, so that new rollers 
must be continually pushed under 
it in front, for the points of con- 
tact A and B between the rollers and the body A B move exactly 
as much backward, in consequence of the rolling, as the axes of 
the rollers move forward. If the roller ^ ZT has turned an arc 
A 0, it has also moved forward the space A Ax equal to this arc, 
O has come in contact with 0,, and the new point of contact 0, 
has gone backward behind the former one (^) a distance A 0, = 
A 0. If we designate the coefficients of friction on H K and A B 
by/ and /i, we have for the force necessary to move the body forward 

-p = (/ + /.) 2^- ■ ''' ■ 




356 GENERAL PRINCIPLES OF MECHANICS. [§ 193. 

Remaek. — The extensive experiments of Morin upon the resistance of 
wagons on roads confirm this law, according to which this resistance in- 
creases directly as the pressure and inversely as the thickness of the rollers. 
Another French engineer, Dupuit, on the contrary, infers from his experi- 
ments, that rolling friction increases directly as the pressure and inversely 
as the square root of the radius of the rollers. The newer experiments of 
Poiree and Sauvage by means of railroad wagons, also lead to the conclu- 
sion, that rolling friction increases inversely as the square root of the radius 
of the wheel. See Comptes rendues de la societe des ingenieurs civils 4 
Paris, 5 et 6 annee. Particular theoretical views upon the subject of roll- 
ing friction are to be found in Von Gerstner's Mechanics, Vol. I, § 537, and 
in Brix's treatise on friction, Art. 6. This subject will be treated with 
more detail in the Third Part, under the head of transportation on roads 
and railroads. 

§ 193. Friction of Cords. — We have now to study the fric- 
„ ^ tion of flexible bodies. If a perfectly- 

flexible cord stretched by a force Q is 
laid over the edge (7 of a rigid body 
ABU, Fig. 292, and is thus compelled 
to deviate from its original direction an 
angle D C B = a°, a pressure E is pro- 
duced at this edge^ which gives rise to a 
friction F, in consequence of which a 
force P, which is either greater or less 
than Q, is necessary to produce unstable 
equilibrium. The pressure is (§ 77) 




R = V P^ + Q^ — 2 P Q cos, a, and consequently the friction 

F = (p VP' -{- Q' - 2 P Qcos.a. 

If now we substitute P = F -\- Q and P" approximatively 
= Q' ']- 2 Q F, we obtain 

F= cj) V Q' + 2 Q F + Q' - 2 Q' cos. a - 2 F Q cos. a 



a 



= 4/2 (1 - COS. a) (Q' + Q F) = 2 4> sin. ■^VQ'-\- Q F,. 

for which we can write 2 </> sin. ^ ( 6 + i F)> when we take into 

account only the first two members of the square root. Hence we 
have 

F = (f> F sin. ^ + 2 (j) Q sin. ^, 

and consequently the friction required is 



§193.] RESISTANCE OF FRICTION, ETC. 357 



F^ 



_ . a 
2 (p Q sm. - 



a' 



1 — ^ sm. ^ 



for which we can generally write accurately enough 

F = 2 (j) Q sin. ~ (1 + <^ sin. ^ j, and very often 

F = 2^ Q sin. ^, 

when the angle of deviation a is very small. Hence, in order to 
draw the rope over the edge C, we need a force 

(2 (j) sin. - \ 
1 + ; —aft 
1 -(psm.-J 

and, on the contrary, the force necessary to prevent the weight Q 
from sinking is 

a 

2 (p sm. - 

P,= Q:\1 + 



1—0 sm. ^ 
we can put approximatively 

P = h -f- 2 sin. ^ ( 1 + si7i. |)J Q, or more simply 

P = (1 +2 (l>sin.^] §and 

F, = ? , or 

a (^ . a\ 

1 + 20 sin. -1 + sin. ^1 

P, = ^ ^ = (1 - 3 <A sin. I) G. 

1 + 20 5W. ^ 

If the cord passes over several edges, the forces P and P^ at the 
other end of the cord can be calculated by repeated application of 
these formulas. Let us consider the simple case, where the cord 
ABC, Fig. 293, is laid upon a body with n edges, and where the 
deviation at each edge is the same and equal to a. The tension of 
the first portion of the cord is 

Q,= (1 +2 05m.^j ft 
when that at the end is — Q; that of the second is 



358 



GENERAL PRINCIPLES OF MECHANICS. 



[§193. 



ft = (l + 2 sin.~] Q,= (l -^ 2(j> sin.^y Q, 
that of the third is 

ft = (l + 2 sin^-'j ft =^1 + 2 </) sin.~y ft 

and in general the tension at the other end is 

P = (l + 2 sin. ^Jq, 

when it is required to produce motion in the direction of the force 
.-^- . I^*®^changing F and ft we obtain the force necessary to pre- 
vent motion in the direction of the force Q and it is 

Q 



P. = 



(l + 2 sin. ^) 



Fig. 293. 



Fig. 294. 





The friction in the first case is 

F= P~Q = [[l + 2 ^ sin. 1)"- l] ft 
and in the second 
F= Q 



P, = [(l + 20 sin. I)' 
= [l - (l + 20sm.0 "J ft 



A 



The same formulas are also applicable to the case of a body 
composed of links, as, e.g., a chain ABE, Fig. 294, which is 
passed round a cylindrical body, when n is the number of links 
lying upon the body. If the length of one joint of the chain is 
= I and the distance C A of the axis ^ of a link from the centre 



§ 194.] 



RESISTANCE OF FRICTION, ETC. 



359 



C of the arc, which is covered, = r, we have for the angle of devia- 

" ' n 7 

tion D B L = A C B = a, sin. ^r = x—* 

2 2 r 

Example. — How great is the friction on the circumference of u wheel 
4 feet high, covered with twenty links of a chain, each five inches long 
and 1 inch thick, when one of the ends is fastened and the other subjected 
to a strain of 50 pounds ? Here we have 

. a 5 5 

Pj = 50 pounds, n = 20, sm. 



2 



49' 



48 + 1 

now if we substitute for ^ the mean value, 0,35, we obtain the friction, with 
which the chain opposes the revolution of the wheel 






1 + 2 . 0,35 



35 V' 
^+ 49o) 



20 
'^] - 1 



50 = 2,974 . 50 = 149 pounds. 



G 



50 



§ 194. If a stretched cord A B, Fig. 295, lies upon a fixed 
tcylindrically rounded body ^ C B, the friction can also be found 

by the rule given in the foregoing paragraph. 
Here -the angle of deviation is B D B = a'' = 
Lingle^ at the centre A C B of the arc A B of the 
cord ; if we divide the same in n equal parts and 
reg-ard the arc A B as consisting of n straight 



Fig. ?9^. 




a. 



lines, we obtain n edges with the deviation — , 



n 



and therefore the equation between the power 
and the load is as in the foregoing paragraph 



-{ 



P = ( 1 + 2 sin. 



a 

¥n 



)"e. 



On account of the smallness of the arc -, sin. ^ — can be re- 

n 2 n 

placed by - — , and we can put 



" - (• - ^j«- 



Developing according to the binomial theorem, we obtain 



P 



l+n 



(pa n(n - 1) (0 af n{n-l) (n-2){(t)ay 



n '^ 1 .2 n' '^ 1.2.3 n^ 



•••)e 



or, since n is very great and we can put n — 1= n — 2 = n — ?> , . . — n. 



360 GENERAL PRINCIPLES OF MECHANICS. [§194, 



x' X" 



But 1 + X -^ tj — ^ + r — ^ — X 4- ... = e"", e being the base 

2,71828 of the Naperian system of logarithms (see Introduction to 
the Calculus), and we can therefore write 

F = e^ ^ , Q OT Q = P e~ ^ ^, and inversely 
1 . P 2,3026 n j^ 7 n\ 

If the arc of the cord is not given in parts of n, but in degrees, 
then we must substitute a = --— . tt, and if finally it is expressed 
by the number u of coils of the rope, we must put a ^= % tt u. 

The formula P = e^ ^ . Q shows, that the friction of a cord 
F — P — § on a fixed cylinder does not depend at all upon the 
diameter of the same, but upon the number of coils of the cord, 
and also that it can easily be increased to almost infinity. If we 
put <^ = ^, we have 

for 1 coils, P = 1,69 Q 

« i " P = 2,85 Q 

"1 " P = 8,12 Q 

« 2 '' P = 65,94 Q 

« 4 " P = 4348,56 Q. 

(Remark.) — From the equation P = (1 + 3 ^ sin. x) Q in § 193, it fol- 
lows tliat 

P-Q = 2<l> sin. I ^, 

or substituting instead of a the element ^ a of the arc and instead of P — Q, 

the corresponding increase <^ P of the variable tension P of the cord and 

putting ^ = P, we obtain 

n ^ r. d a. ^ d P , 

^P = 3 — P, or -^ = ^ 6? a, 

whence by integration we obtain 

I P = ^ a + Con. 
In the beginning a is = and P = Q, and therefore we have 

P 

I Q = + Con. and lP--lQ = lj- — ^a^ 

or inversely 

— = e"^ ,orP=6^ Q. 



§ 195.] 



RESISTANCE OF FRICTION, ETC. 



361 



Example. 

Fig. 296. 



In order to let down a shaft a very great but indivisible 
weight P = 1200 pounds, we wind the 
rope, to which this weight is attached, 
If times around a firmly fastened log 
A B, Fig. 296, and we hold the other 
end of the rope in the hand. What force 
must be exerted at this end of the rope, 
when we wish the weight to descend 
slowly and uniformly ? If we put here 
(j) = 0,3, we obtain for this force 




1200 . e 



-0,3 



= 1200 . e 



3i 



33 



I q^l 1200 - — 7T = 7,0901 - 2,5918 

= 4,4983, 
or log Q = 1,9536, whence 
Q = 89,9 pounds. 



§ 195. Rigidity of Chains. — If ropes or bodies composed of 
links, etc., are laid on a pulley or a cylinder movable about its 
axis, the friction of cords and chains considered in the last para- 
graph ceases, because the circumference of the wheel and the cord 
have the same velocity, and hence force is only necessary to bend 
the rope as it lays itself upon the pulley, and sometimes to 
straighten it as it is unrolled from the pulley. 

If it is a chain, which winds itself around a drum, the resistance 
during the rolling and unrolling consists of the friction of the bolts 

against the links, since the 
former are turned through a 
certain angle in their bear- 
ings. If A B, Fig. 297, is a 
link of the chain and B G 
the following one, if C is the 
axis of rotation of the pulley, 
upon which the chain, stretch- 
ed by the weight Q, winds, 
and if finally (7 if and (7 iV^ 
are perpendiculars let fall 
upon the major axis of the 
links A B and B G, then 
M C N = a° is the angle 



Fig. 297. 




362 



GENERAL PRINCIPLES OF MECHANICS. 



[^ 195. 



turned througli by the pulley, while a new link lays itself upon it, 
and KB G = 1S0° — A B G is the angle described by the link 
B G with its bolt B D upon the link A B during the same time. 
If B D = B E = r^ is the radius of the bearing of the bolt, the 
point i) of the pressure or friction describes an arc D E =^ Ti a, 
while a link lays itself upon the roller, and the work done by the 
friction at the point D is, = 0i Q . Tj a. Supposing the force Pj 
necessary to overcome this friction to act in the direction of the 
greater axis B G, we have the space described by it in the same 
time s = (7 iV^ multiplied by the arc of the angle M G N=C N .a, 
and therefore the work done = Pi . G N . ct, equating the two 
mechanical effects, we have Pi , G N , a = <i>i . Q ri a, and the force 
required is 

a denoting (7iV^the radius of the drum plus half the thickness of 
the chain. 

If we neglect the friction, the force necessary to turn the 
pulley would be P = (g, 

but when we take into account the friction caused by the winding 
of the chain upon the pulley, we have 

p = e + Pi = (i + 01 ™) e. 

If the chain unwinds from the drum, the resistance is the same ; 
if, therefore, as on a fixed pulley, the rope is wound upon one side 
and unwound upon the other, the required force is 

P =: (l + 0j —1 Q^ or approximatively — (1 + 2 01 — j Q. 

If, finally, the pressure on the axle is = P and the radius of the 
axle — r, the force necessary to overcome all the resistances is. 



P = (l+20,^)? 



4- 0— P. 
a 



Example. — How great is the force P at the end of a chain passing 



Fig. 298. 




round a roller A G B, Fig. 298, when the weight 
acting vertically is Q = 110 pounds, the weight 
of the roller and chain is 50 pounds, the radius a 
of the roller, measured to the middle of the chain, 
is <x = 7 inches, the radius of the axle (7 is = f of an 
inch and that of the bolts of the chain is = f of an 
inch ? If we put <p — 0,075 and ^^ = 0,15, we obtain, 
according to the last formula, the force 

P={\+^. 0,15 . -|^yil0 + 0,075.^ (110 + 50 + P), 



§ 190.] 



RESISTANCE OF FRICTION, ETC. 



363 



or assuming in the right-hand member P appro ximatively = 110 
P = 1,016 . 110 + 0,0067 . 370 = 111,76 + 1,81 = 113,6 pounds. 



Fig. 299. 



§ 196. Rigidity of Cordage- — If a rope is passed oyer a pulley 
or winds itself upon a shaft, its rigidity (Fr. roideur, Ger. Steifig- 
keit) comes into play as a resistance to its motion. The resistance 
is not only dependent upon the material, of which the rope is made, 
but also upon the manner, in which it is put together, and upon the 
thickness of the rope ; it can consequently be determined by experi- 
ment alone. 

The principal experiments for this object are those made by 
Coulomb and those made more recently by the author himself. 
While Coulomb employed only small hemp ropes from | to at most 
1^ inches in thickness and made them wind upon rollers of 1 to at 
most 6 inches in diameter, the author employed hemp ropes 2 
inches thick and wire ropes from ^ to 1 inch thick and passed 
them over rollers from 2 to 6 A feet in diameter. Coulomb's experi- 
ments were made in two different ways. In 
one case, like Amonton, he employed the 
apparatus represented in Fig. 299, where A B 
is a roller around which two ropes are wound, 
the tension being produced by a weight 
and the rolling down of this roller by a weight 
jP, which pulls upon this roller by means of a 
thin string. In the other case he laid the 
ropes around a cylinder rolling upon a hori- 
zontal surface and, after having subtracted the 
rolling friction, calculated the resistance of the 
rigidity from the difference of the weights, 
which were suspended to. the two ends of the 
rope and which produced a slow rolling motion. 

According to the experiments of Coulomb, the resistance of the 
rigidity increases tolerably regularly with the amount of the ten- 
sion of the rope; but there is also a constant member K, as might 
have been expected; for a certain force is necessary to bend an un- 
stretched rope. It was also shown, that this resistance was inversely 
proportional to the radius of the roller; that for a roller of twice 
the diameter it is only one-half, for one of three times ihe diam- 
eter, one-third, etc. Finally, the relation between the thickness 
and rigidity of a rope can only be determined approximatively from 
these experiments, as we might have supposed ; for this rigidity de- 




Q 

ililM.I 

i[i)iilihiiliilinlM 'l! 



364 GENERAL PRINCIPLES OF MECHANICS. [§ 197. 

pends upon the nature of the material of the ropes and upon the 
size of the fibres and strands. When a rope is new, the rigidity is pro- 
portional, approximatively, to d^'\ and when it is old, to d^'*, d 
denoting the diameter of the rope. The assumption by some 
authors that it varies with the first power, and that of others that 
it varies with the square of the thickness of the rope, are therefore 
only approximatively true. 

§ 197. Prony's Formula for the Rigidity of Hemp 
Hopes. — According to the last paragraph, the rigidity of hemp 
ropes can be expressed by the following formula : 

in which d denotes the thickness of the rope, a the radius of the 
pulley measured to the axis of the rope, Q the tension of the rope, 
which passes round the pulley, and n, K and v empirical con- 
stants. Prony found from Coulomb's experiments for new ropes 



and for old ones 



8^~ (2,45 + 0,053 q\ 



8, = — (2,45 + 0,053 0, 

Cv 



in which formulas a and d are expressed in lines and Q and 8 in 
pounds. These formulas are, however, based upon Paris measures ; 
for English measures they become, when expressed in inches and 
pounds, ^ ^ ^' ^^^^33 ^ ^^^gg ^^ 

8, = ~ (6,96 + 0,14 Q). 

Since even these complicated formulas do not agree as well as 
could be wished with the results of experiment, we can, as long as 
we do not take into account the later experiments, write with 
Eytelwein 

^ ^» d"" Q 



a ^ 3604^* 

In this formula a must be expressed in English feet and d in 
English lines, but Q and ^S' may be expressed in any arbitrary sys- 
tem of weights. If we employ the metrical system of measures, 
we have 



§197.] RESISTANCE OF FRICTION, ETC. 365 



S = 18,6 . 



a 



The results given by this formula are not sufficiently accurate, ex- 
cept when the tension upon the rope, as is generally the case in 
practice, is very great. 

The rigidity of tarred ropes was found to be about one-sixth 
greater than that of untarred ones, and wet ropes were found to be 
about one-twelfth more rigid than dry ones. 

Example. — If the tension upon a new rope 9 lines thick, which passes 
round a pulley 5 inches diameter, is 350 pounds, the rigidity, according to 
Prony, is 

8 = f (1)^'' (14,39 + 0,389 . 350) = 0,613 . 46,216 = 28,33 pounds, 
and according to Eytelwein 

„ 9''. 350 ^^_ 

^ = 3604-^=^^'^^^"^^^^- 

K the tension were but Q = 150 pounds, we would have, according to 
Prony, 

S = 0,613 . 23,1 = 14,16, 

and according to Eytelwein 

cr 81 . 150 

^ - 36047^ - ^®'^- 

In this case the formulas give results, which coincide better with each 
other. "We see from the above example, how uncertain these formulas are. 

Bemabk. — Tables for facilitating the calculation of the resistance due 
to the rigidity of cordage will be found in the Ingenieur, page 365. Ac- 
cording to Morin (see his Lecons de Mecanique Pratique), we have, when 
n denotes the number of strands in the rope and a the radius of the pulley 
in centimetres, for untarred ropes 



d = V0,13387i centimetres and 

S = ^ (0,0297 + 0,0245 n + 0,0363 Q) kilograms 

= — (0,1110 + 0,6843 d^ + 0,1357 Q) kilograms, 

and for tarred ropes 

d = V 0,186 7i centimetres and 

5 = x^ (0,14575 4- 0,0346 n + 0,0418 Q) kilograms 

d? 

= — (0,3918 -f 0,5001 d'' -f 0,1124 Q) kilograms. 



3CG 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 198. 



If, however, d and a are expressed in inches, and 8 and Q in pounds, 
we can put for untarred ropes 

d'' 
S = ^ (0,621 + 34,70 d' + 0,3445 Q), 

and for tarred ones 

d^ 
8== — (2,193 + 18,06 d^ + 0,2889 Q). 

If, E.G., for an untarred rope we have ^ = f inch, <» = f inches and 
Q = 350 pounds, then 



9 2 



9 



Fig. 300. 

D 



S = —- (0,621 + 24,70 . T7J + 0,3445 . 350) 
lb 5 lb 

9 

= — (0,621 + 13,893 + 120,575) = 30,4 pounds, 

while in this case (last example) Prony's formula gave 8 = 28,33 pounds. 

§ 198. Experiments Upon the Rigidity of Thick Ropes. — 

The author, in his experiments upon the rigidity of cordage, made 
use of the apparatus represented in Fig. 300. The sheave or roller 

B D E, over which the rope to be 
tested is passed, was, together with a 
pair of iron wheels C L M, fastened 
upon a shaft or axle C, and these 
wheels ran upon two horizontal rails 

I IH|HWH|Hi" ^ ^' "^^ ^^® ^^^ ^ ^^ ^^^®- rope a 
Y |^\ \^M<r|,# %| weight G was attached, and to the 

other end A a cross K, upon which, 
weights were hung until the wheels 
and pulley began to roll forward 
slowly. In order to be as independ- 
ent as possible of errors arising from 
imperfections in the apparatus, addi- 
tional weights were afterwards added 
at F until a rolling motion in the 
opposite direction was produced. The 
arithmetical mean of the weights 
added gave, when, the rolling fric- 
tion was deducted, the rigidity of 
the rope. The coefficient ^f rolling 
friction to be used was determined in the same way, except that a 
thin spring, whose rigidity could be neglected, was employed instead 
of a rope. The mean value of this coefficient was given in § 192. 




g 199.] RESISTANCE TO FRICTION, ETC. 367 

The resistance due to the rigidity is, according to the author's 
views, due less to the rigidity proper than to the friction of the 
different wires or strands upon each other ; for in passing over the 
pulley, they naturally change their relative positions. When a 
wire rope passes round a fixed pulley, the first part of this resist- 
ance is wanting, as the rope, in consequence of its elasticity, gives 
out, when it straightens itself, as much mechanical effect as was em- 
ployed in bending it around the pulley. Hence the rigidity of the 
rope in this case consists solely of the friction of the wires upon 
one another, a conclusion which is confirmed by the author's ex- 
periments ; for he found the resistance to be forty per cent, less, 
when the ropes were freshly oiled or tarred than when they were 
dry. The conditions are different in the case of hemp ropes, for 
they do not possess, especially after long use, any elasticity, and 
the strands and fibres require force not only to bend them, but also 
to straighten them. 

§ 199. New Formulas for the Resistance Due to the 
Rigidity of Cordage. — Since the rigidity of a rope depends not 
only upon its thickness, but also upon the amount of bending it is 
subjected to, and also upon the manner in which it is put together, 
the author considers, that these conditions can be very well ex- 
pressed by the formula 

a 
the constants Xand v must be determined specially for each kind 
of rope. The experiments of the author also showed, that for wire 

K 

ropes we should put simply K instead of — , or 



8 = K + 



a 
vQ 



a 

1. For tarred hemp ropes 1,6 inches thick passing round sheaves 
from 4 to 6 feet in diameter, he found 

^S' = 1,5 + 0,00565 -^ kilograms, 
when the radius a is expressed in metres, or 
8 = 3,31 + 0,222 ^ pounds, 

when a is expressed in inches. 

2. For a new untarred hemp rope | inch thick, upon a pulley 
21 inches in diameter, he found 



368 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 200. 



s 



0,086 + 0,00164-? kilograms = 0,1896 + 0,06457 -? pounds. 



3. A wire rope 8 lines in diameter, formed of 16 wires, each 1^ 
lines thick, and weighing 0,68 pound per running foot, was passed 
around pulleys from 4 to 6 feet in diameter, and gave 

S = 0,49 + 0,00238 -? kilograms = 1,08 + 0,0937 ^ pounds. 

4. For a freshly-tarred wire rope, with a hemp centre in each 
strand and in the rope, which was 7 lines in diameter, was com- 
posed of 4.4 =■ 16 wires, each 1-^ lines thick, and weighed 0,67 
pound per running foot, he found, with a pulley 21 inches in 
diameter, 

B - 0,57 4- 0,000694 ^ kilograms = 1,26 4- 0,0272 -^ pounds. 

Remark, — A detailed description of the author's experiments is to be 
found in the Zeitschrift fur Ingenieurwesen (dem Ingenieur), by Borne- 
mann, Briickmann and Roting, Vol. I, Freiberg, 1848. The hemp ropes 
of 1 were formerly employed in Freiberg for hoisting from the shafts by 
means of a water-wheel and drum (Ger. Wassergopel), but of late they 
have been replaced by the wire ropes of 3 and 4. Both of these kinds of 
ropes can support with sextuple security a load of 80 cwt. It was shown 
by the above experiments that, when the load was the same, the resistance 
due to the rigidity of wire ropes was less than that due to the rigidity of 
hemp ones. If we assume the tension of the rope to be Q = 2000, and the 
radius of the sheave to be a = 40 inches, we have for hemp ropes 

8 = 3,31 + 0,322 ^^ = 14,41 pounds, 
and, on the contrary, for wire ropes 

S =1,08 + 0,0987 -2^^ = 5,8 pounds. 

§ 200. Theory of the Fixed Pulley. — Let us now apply 
the principles just enunciated to the theory of the fixed pulley. 



Fig. 301. 



Fig. 802. 





Let A C B, Fig. 301 or Fig. 302, be the pulley, and let a be its 



§200.] RESISTANCE TO FRICTION, ETC 369 

radius C A = C B, r the radius of its axle, G its weight, d the 
thickness of the rope, Q the weight suspended to one end of the 
latter, S the resistance due to the rigidity, F the friction upon the 
axle, reduced to the circumference, and F = Q -{- F -\- S the force 
at the other end of the rope. The rigidity of the rope is shown by 
the fact that the rope does not immediately assume the curvature 
of the pulley as it is wound upon the sheave, nor straighten itself 
immediately, when it is unwound. On the contrary, it approaches 
the sheave in an arc, the curvature of which constantly increases, 
and leaves in an arc, the curvature of which constantly diminishes. 
The difference between the elastic wire ropes and the unelastic 
hemp ones is that the former leave the sheave somewhat sooner 
and the latter somewhat later ; hence the arm C D of the force in 
the first case (Fig. 301) is somewhat greater, and in the second case 
(Fig. 302) somewhat less than the radius O A = a of the sheave. 
If we neglect the friction upon the axle and put F = [Q + S), 
we have 

(Q + S).CD= Q. GE, 

and consequently the rigidity of the rope is 



S 



and the ratio of the arms is 

GE _ 8_ 

GD~ ^ Q' 
the value of which can easily be calculated by substituting one of 
the values of S. 

We can also determine this force F = Q ■{■ 8 -{- F without 
employing the ratio of the arms of the lever by substituting in 
that formula either with Prony for thin hemp ropes 

8=^{K+vQ), 

or with the author for wire or thick hemp ropes. 

a 

and the friction upon the axles reduced to the circumference of 
the pulley is 

T 

F=(t>-{Q-hG + F), or approximatively. 



a 



F=(t>-{2Q+ G). 



24 



370 GENERAL PRINCIPLES OP MECHANICS. [§200. 

Hence, in the first case, we have 

and in the second 

In the case of the wheel and axle a reduction of the force from 
the circumference of the axle to that of the wheel is necessary. 

Example. — If a wire rope 8 lines in diameter passes over a pulley 
5 feet high, whose axles are 3 inches in diameter, and if the tension upon 
the rope is 1300 pounds, we have the required force, when the coefficient 
of friction is ^ = 0,075 and the weight of the pulley = 1500 pounds 

P ==. 1200 + 1,08 + 0,0937 . ^%^ + 0,075 . -^ (2400 + 1500) 
= 1200 + 1,08 + 3,748 + 14,62 = 1219 pounds; 

hence i| = 1,6 per cent, of the force is lost in consequence of the rope's 
passing round the pulley. 

If instead of a wire rope we employed a hemp one 1,6 inches thick, we 
would have 

P = 1200 + 3,31 + 0,222 . iffa + 14,63 = 1337 
and the loss of force would be 

37 
P — Q = —- = 3,25 per cent. 



FOURTH SECTION. 

THE APPLICATION OF STATICS TO THE ELAS- 
TICITY AND STRENGTH OF BODIES. 



CHAPTER I. 

ELASTICITY AND STRENGTH OF EXTENSION, COMPRESSION 

AND SHEARING. 

§ 201. Elasticity. — The molecules or parts of a solid or rigid 
body are held together by a certain force, called cohesion (Fr. cohe- 
sion; Ger. Cohiision), which must be overcome, when the body 
changes its form and size, or if it is divided. The first effect, which 
forces produce upon a body, is a variation in the relative position 
of its parts, in consequence of which a change of form and volume 
occurs. If the forces acting upon a body exceed certain limits, a 
separation of the parts takes place and perhaps a division of the 
whole body into pieces. The capability of a body to resume its 
original form, after the force which caused its change of shape has 
been removed, is called in the most general sense of the word its 
elasticity (Fr. elasticite ; Ger. Elasticitat). The elasticity of every 
body has certain limits. If the change of form and volume exceeds 
a certain amount, the body remains of the same form after such a 
change, although the forces which have produced the variation 
have ceased to act. The limit of elasticity is very different for 
different bodies. The bodies, which permit a great change of 
volume before their limit of elasticity is reached, are cdWoi^ perfectly 
elastic ; those, whose limit of elasticity is reached when they liave 
undergone a very slight change of form, are called inelastic, 



372 GENERAL PRINCIPLES OF MECHANICS. [§203. 

although no such bodies really exist. It is an important rule in 
architecture and in the construction of machinery, not to load the 
materials employed to such an extent that the change of form 
produced shall reach, much less exceed, the limit of elasticity. 

§ 202. Elasticity and Strength.— Different bodies present 
different phenomena, when they are changed in their form beyond 
the limit of elasticity. If a body is hrittle (Fr. cassant ; Ger. sprode),, 
it flies in pieces, when its form is changed beyond its limit of elas- 
ticity ; if, however, it is ductile or malleaUe (Fr. ductile ; G-er. ge- 
schmeidig), as, E.G., many metals, we can cause considerable 
changes in its form beyond its limit of elasticity, without causing 
a separation of its parts. Some bodies are hard (Fr. dur ; Ger. hart), 
others soft (Fr. mou ; Ger. weich) ; while the former oppose great 
resistance to a separation of their parts, the latter permit it with- 
out much difficulty. 

We understand by elasticity, in the more restricted sense of the 
word, the resistance with which a body opposes a change of its 
form, and by strength (Fr. resistance, Ger. Festigkeit) the resistance 
with which a body opposes division. In what follows, both sub- 
jects will be treated. According to the manner in which the extra- 
neous forces act upon bodies, we can divide elasticity and strength 
into 

I. 8imple and 

II. ComUned; 

and the former again into 

1) Absolute or the elasticity and strength of extension, 

2) Reacting, or the elasticity and strength of compression, 

3) Relative, or the elasticity and strength of flexure, 

4) The elasticity and strength of sheering and 

5) The elasticity and strength of torsion or twisting. 

If two extraneous forces P and — P act by extension (Fr- 

traction, Ger. Zug) in the direction of the axis of a body A By Fig. 

Fig. 303. 303, the latter resists the 

-^' '^ means of its absolute elas- 

ticity and strength or its elasticity and strength of extension (Fr. 
elasticite et resistance de traction, Ger. Zug oder absolute Elasticitat 



§ 202.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 



373 



und Festigkeit) ; if, on the contrary, two forces P and — P press 
Fig. 304. the body together in the direction 

of the axis of the body A B, Fig. 

304, so that the latter is compressed 
and finally crushed, the elasticity and strength of compression or the 
reacting elasticity and strength (Fr. elasticite et resistance de com- 
pression, Ger. Druck or riickwirkende Elasticitat und Festigkeit) 
must be overcome. If, farther, three forces P, Q, R, which balance 
each other, are applied at three different points A, B, C, in the 
axis of the body A B, Fig. 305, and act at right angles to the same, 
this body would be bent or perhaps broken, and it is the relative 
elasticity and strength, or the elasticity and strength of flexure (Fr. 
elasticite et resistance de flexion, Ger. Biegungs oder relative Elas- 
ticitat und Festigkeit), that must be overcome, in order to bend 
or break it. If, in the latter case, the points of application A and 
C He close together, as is represented in Fig. 305, a distortion is 



Fig. 305. 



Fig. 306. 





produced in the cross section D D, between the two points A and 
C ; if the force P is great enough, the body is divided into two 
parts, and in this case the elasticity and strength of sheering (Fr. 
elasticite et resistance par glissement cisaillement ou tranchant, 
Ger. Elasticitat und Festigkeit des Abschierens) is overcome. If 
tw^o couples (P, — P), {Q, — Q), which balance each other, act upon 
a body C A, Fig. 306, in such a manner that their planes are at 
right angles to the axis of the body, a ticisting of the body is pro- 
duced, which may become a wrenching, and here the elasticity and 
strength of torsioii (Fr. elasticite et resistance de torsion, Ger. Dreh- 
ungs-elasticitat und Festigkeit) is to be overcome. 

If several of the forces here enumerated act at the same time 
upon a body, the combined elasticity and strength or a combination 
of two or more of the simple elasticities and strengths comes into 
play. 



374 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 203. 



Fm. 307. 




Fig. 308. 
P 




§ 203. Extension and Compression.— The most simple 
case of elasticity and strength is presented by the extension and 
compression of prismatic bodies, when they are acted npon by 
forces whose directions coincide with the axis of these bodies. It is 

of course not necessary 
that both should be 
motive forces. The ac- 
tion is the same, when 
the body is firmly sus- 
pended or supported at 
one end and at the 
other end subjected to 
a pull or to a thrust. 
We can obtain an ex- 
ample of this case ei- 
ther by suspending to 
a prism A B C D, Fig. 
307, which hangs vertically, a weight P, or by loading with a weight 
P a prism A B C D, Fig. 308, which is supported at the bottom. 
In the first case, the body is extended a certain amount C d = 
D Di — X, and in the second case, it undergoes a similar com23res- 
sion ; if, therefore, the initial length of the body h A D = B C = 
I, it becomes, in the first case, 

A D,=^ B C,:=AD + D D,=l+ X, 

and in the second case, 

A I), = B a = A D - D n, --= I— }.. 

The extension or compression a increases with the pull or thrust 
P, and is a function of the same. This function or algebraical 
relation between P and /I cannot be. determined a 2^Tiori ; it is 
dependent upon the physical properties of the body, and is different 
for different materials. If we regard P and A as the co-ordinates of a 
curve and construct this curve with the corresponding values of P 
and X determined by experiment, we obtain by this means not only 
a graphic representation of the law, according to which bodies are 
extended and compressed by extraneous forces, but also a means of 
determining the peculiarities of this law. 

If we lay off from A on the positive side of the axis X X, 
Fig. 309, the tensions or tensile forces, which act upon a body, as 
abscissas A B, A M, etc., and at their ends the corresponding 



§ 203.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 375 

extensions as ordinates B D, M 0, etc., parallel to Y Y, we obtain 
a curve A D W, which represents the law of the extension of 




this body ; and if, on the contrary, we cut off on the negative side 

of the axis X X from A the pressures or thrusts as abscissas A Bx, 
A Ml, etc, and at the extremity of the same lay off the correspond- 
ing compressions as ordinates B^ D^, M^ Oi, etc, we o])tain a curve 
A />! Oi Wi, by which the law of compression of the body is graph- 
ically represented. According to the results of many experiments, 
these two curves pass without interruption into one another, have 
consequently at .^ a common tangent G A G^, and are therefore 
properly only branches of the same curved line W D A D^ Oi Wj. 
Although the curve as a whole differs considerably from a right 
line, yet in the neigborhood of the origin of co-ordinates A it 
nearly coincides with the tangent G A G^, and since for this line 
the ordinates are proportional to the abscissas, we can also assume 
that the small extensions and compressions produced hy the pulls or 
thrusts A B, A B^, etc., are proportional to these forces (Hooks' 
Law). 

The total extension M 0, produced by the pull A M, consists 
of two parts, viz.: the permanent extension or set M Q, which 
remains in the body, when the stress has ceased to act, and the 
elastic extension Q 0, which vanishes with the pull. It is the same 
for compression. The total compression My 0, is the sum M^ Qx + 



376 GENERAL PRINCIPLES OF MECHANICS. [§304. 

Qi Ox of the permanent compression or set M^ Q^ and of the elastic 
one Qi Ox. When the forces are small, the permanent change is so 
very small compared with the total one, that it can be regarded as 
not existing, and consequently the total extensions and compres- 
sions can be treated as the elastic ones. If the force exceeds a cer- 
tain limit A B {A Bx), the so-called limit of elasticity, if, e.g., it 
becomes A M {A i/i), the permanent change of length or set forms 
a considerable portion of the total extension Jf or of the total 
compression Mx Ox, If the pull or thrust reaches a certain value 
A U or A Ux, the extensions U R, U PFand the compressions Ux Ri 
and Ux Wx attain the limit at which the cohesive force of the body 
is no longer able to balance the pull or thrust, and consequently a 
tearing asunder or a crushing of the body takes place. 

If a body has been subjected to a force, which has not extended 
or compressed it beyond the limit of elasticity, the body will not 
assume any further set, when subjected to another pull or thrust, 
which does not reach the limit of elasticity. 

§ 204. Fundamental Laws of Elasticity. Modulus of 
Ijlasticity. — The lengthening or extension of a prismatical body, 
produced by a force P, is proportional, in the first place, to the 
length I of the body, since we can assume that equally long por- 
tions are equally extended, and it is inversely proportional to the 

cross-section F of the body, since we can sup- 
FiG. 310. pQgg ^]^g entire stretching force to be equally dis- 

'^ ^ tributed over the entire cross-section of the body. 

If, therefore, a body A B, Fig. 310, whose length 
is = unity and whose cross-section = unity, is 
extended an amount a by a stress P, the exten- 
sion produced in another body F G of the same 
material, whose length is — I and whose cross- 
section is = F, by the same stress is 

_ <jI 

The extension g is of course dependent upon 
the pull P alone and is different for different 
^ materials ; but according to what precedes 
(§ 203) we can assume that for small pulls, which do not exceed 
the limits of elasticity, the extension is proportional to the cor- 
responding stress, or that the quotient ^ is a constant quantity. 



6 
B^ 



§204.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 377 

Now if A B, Fig. 311, represents the tension P of a prism, 
whose length is = unity and whose cross-section = unity, within 




the limits of elasticity and B D the corresponding extension o, 
and if we denote the angle G A U = D A B of the tangent to the 
curve of extension at A by a, we have also 

tang, a = — — , and therefore 

a =z P tang, a, whence we obtain 
^ . . PI tana, a 

1) ^ = — jf^. 

The quantity tang, a is dependent upon the physical proper- 
ties of the body and can be determined by experiment only. If 
we assume I = 1, F = 1 and P = 1, we obtain tang, a — X, and 
this quantity tang, a, to he determined hy experiment, is the exten- 
sion which is produced in a prism, tvhose length is unity and whose 
cross-section is unity, hy the tensile force unity (see Combes : Traite 
de V exploitation des mines, tome I.). If in the formula (1) we 
assume F = 1 and A == ?, we obtain the expression 

1 = P tang, a, or = cotang. a = P; 

-I 

is that force, which would stretch a prism, whose cross* 



hence 

tang, a 

section is one square inch (1), its own length, were that possible with' 

out surpassing the limit of elasticity. 



378 GENERAL PRINCIPLES OF MECHANICS. [§204. 



This hypothetical empirical quantity =: cotg. a is called 

the modulus of elasticity (Fr. coefficient d'elasticite ; Ger. Elastici- 
tatsmodul) of the body or material and will hereafter be designated 
by the letter E. 

According to this we have 

^)^ = ^. 

or the relative extension, i.e., its ratio to the entire length of the 
body 3) - = ^ 



I ~~ F E" 

Inversely the force corresponding to the extension A is 

4) P = jFE, 

The same formulas obtain also for the compression A, caused by 
a tlirust P, and the modulus of elasticity E = cotang, a is the same 
as for extension as long as the limit of elasticity is not sur- 
passed, although in this case it denotes that force, which would 
compress a prism of the cross-section unity its whole length, or to 
an infinitely thin plate, provided that this were possible without 
exceeding the limits of elasticity. 

Remark 1. — We can also put the modulus of elasticity E equal to the 

weight of a prism of the same material as the body, upon which E acts, and 

of the same cross-section unity. If a is the length of this body and y the 

heaviness or the weight of one cubic inch of the same material, we have 

E 
E =. a y^ and therefore inversely « = — 

Tredgold (after Young) used this length as the measure of the elasticity 

(see T. Tredgold on the strength of cast iron and other metals). If E is, 

E.G., 30000000 pounds for cast steel and y = 0,3 pounds, we have 

30000000 
a = — ^r^ — = 100000000 inches, 

I.E., a steel rod 100000000 inches long, would extend a steel bar of the same 
cross-section its whole length, if the law of extension given above were true 
for all limits. 

Remark 2. — During the extension or compression of a body a change 
of cross-section takes place, which, according to Wertheim (see Comptes 
rendues, T. 26), amounts to f of the longitudinal extension or compression. 
If I is the initial length, F the initial cross-section and Fthe initial volume 
Floi the body, l^ and F^ being the length and cross-section during the 
action of the force P, we have the corresponding volume 

Fi=Pi h = FI + F{1^ -I) -{F-F^) I, 01 

r^- r= F (i^ -I) - (F- F,) I, 



g 205.] ELASTICITY •AND STRENGTH OF EXTENSION, ETC. ^^^ 



and the relative chang:e of volume is 



But we know that 



V "~ ~1 F~ 

F-F. _ il. - l\ 



F 



' = « (^> 



whence it follows that 

I.E., the incfrease in wlume is one-third the increase in length. 



According to the theory of Poisson, 



V 



--=,f-Vi 



Example — 1) If the modulus of elasticity of brass wire is 14000000 
pounds, what force is necessary to stretch a wire 10 feet long and 3 lines 
thick one line ? Here we have 

Z = 10. 12 = 120 inches, 1 = -jJ^inch and consequently ,- = xsVo ; 

bat F = —T- = 0,7854 (i^)^ = 0,0218 square inches, hence the force re- 
quired is 

p = __i_. 0,0218 . 14000000 = 212 pounds. 

2) If the modulus of elasticity of iron wire is 31000000 pounds, and an 
iron surveyor's chain 66 feet long and 0,2 inch thick is submitted to a pull 
of 150 pounds, the increase in length is 

' = 0,785^(0,2)- • 3W00 = ^'^^^ ^^"^^^ = ^'^^^ ^^^'^• 

§ 205. Proof Load, Proof Strength, Ultimate Strength. — 

The force A B, Fig. 312, which stretches a prismatical body, whose 




380 GENERAL PRINCIPLES OF MECHANICS. [§305. 

cross-section is unity, to the limit of elasticity, is called the modulus 
of proof strength of extension^ and will in future be designated by 
T, while the thrust necessary to compress the same to its limit of 
elasticity is called the modulus of proof strength of compression, and 
will hereafter be designated by T^. 

From the moduli of proof strength I'and T^, with the aid of ^ 
the modulus of elasticity E, the extension g and the compression a, 
at the limit of elasticity can easily be found ; for we have 

- = -^and- = ^. 

If F is the cross section of a prismatical body, whose moduli of 
proof strength are Tand Ti, we have their prdof strength or proof 
load 



^. (for a pull, P = FT 

^ i and for a thrust, F^= F T,. 



In constructions the bodies should never be loaded beyond their 
limit of elasticity, and the loads should therefore never surpass the 
proof strength of the cross-section of the prismatical bodies em- 
ployed. Cross-sections must therefore be determined by the follow- 
ing formulas : 

P 



2 



,F — y= and 



On account of the accidental overloading and concussions, to 
which buildings and machines may be subjected, and also on ac- 
count of the changes, which the bodies undergo in the course of 
time, owing to the action of air, water, etc., we render these con- 
structions safer by substituting in the foregoing formula, instead 
of .the proof load, only one-half or one-third of the same, i.e. by 
making the cross-section two or three times as great as those given 
directly by the formula. In order to have an mfold security, we 

P P 

must substitute in the formulas F = -^ ox Fx = y^, instead of T 

T T 

or Txi the toorhinq or safe loads — or --. 

m, m 

The force A U, Fig. 313, necessary to tear apart a prismatical 

body, whose cross-section is unity, is called its modulus of rupture 

or of ultimate strength of extension, and is denoted by the letter K; 

and in like manner we call the force A Z7i which crushes a body, 
whose cross-section is unity, the modtdus of rupture or of ultimate 



§205.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 381 



strength of compression, and we denote it by Kx. If the cross-sec- 
tion of the prismatical body is F, we have 

Fig. 313. 
Y 




-y-Sl 



3) 



( P = FXfoT the force, which will tear the body, and 



F Kx for the force, which will crush it. 
The cross-section of bodies is often determined from the modu- 
lus of rupture by substituting in the formulas 

P 



4) 



F = ^ and 



i^i = 



^1 



instead of K the worhing load of rupture, i.e. a small part — or — , 

E.G., a fourth, sixth, tenth, etc., of the numbers determined by ex- 
periment. We call n a factor of safety. If the proof strength of 
all substances were the same fraction of the ultimate strength, that 



is, if the ratios 



A B _ T ' ABx 



T 

= —^ were fixed constants, the 



A U K AUx Kx 

determination of the cross-section by means of the moduli of proof 
strength would give the same result as that by means of the work- 
ing load of rupture ; but since this ratio is different for different 
bodies, the determinations by the aid of the moduli of proof 
strengths T and Ti, or rather by means of tlie working or safe loads 

T T 

-— and — , are generally more correct and proper, and the deter- 



382 GENERAL PRINCIPLES OF MECHANICS. [§206. 

mination by tlie tvorhing or safe loads of rupture — and — ^ is only 

to be employed, when the modulus of proof strength is unknown. 
If the cross-section of a body is a circle, whose diameter is d, we 

have -^ = F, whence P = ^V = 0,7854 d' Tand 
4 4 

^ = i/— z= 1,128 V'F :::^ 1,128 \/ ~ 

Example 1. — ^What weight can a hanging column of fir support, if it is 
5 inches wide and 4 inches thick? Assuming the modulus of proof 
strength to be 3000 pounds, the cross-section being i^ = 5 . 4 = 20 square 
mches, we have P = F T = 20 . 3000 = 60000 pounds as the proof load 
of this column. If, however, we assume the modulus of rupture to be 
K = 10000 pounds, and we desire a quadruple security, we have P= FK 
— 20 . iQ^"Q = 50000 pounds. In order to be secure for a great length of 
time, we take but a tenth part of K, and obtain thus P = 20 . 1000 = 
20000 pounds. 

Example 2. — A round wrought-iron rod is to be turned so as to bear a 
weight of 4500 pounds; what should be its diameter? Here T is 18700 

pounds, whence d = 1,128 r Y«7on = ^'^^^ "^ m7 ^ ^'^^^ inches. The 

modulus of rupture of average wrought-iron is = 58000 pounds ; if, how- 
ever, we wish five-fold security, we take K = 11600 pounds, and we have 

d = 1,128 i/^^-r, = 1,128 V^ = 0,7025 inches. 
^ llbOO ' lib 

§ 206. Modulus of Resilience and Fragility. — When we 
stretch a prismatical body by a force, which gradually increases from 
to F = A M = li 0, Fig. 314, and by this means lengthen it 
from to A = if = ^ iV, a certain amount of work is done, 
which is determined by the product of the space or total extension 
A JV and the mean value of the pull, which increases gradually from 
to P = iV^ 0. This product can be expressed by the surface 
A N 0, whose abscissa is the extension ^ iV = A and whose ordi- 
nate is the pulling stress JV = A M = P. If the extension does 
not exceed the limit of elasticity, the surface A JV can be con- 
sidered as a right-angle triangle, whose base and altitude are A and 
P, and the work done, corresponding to it, is 

Z = i A P. 

If we substitute in it 

A = (7 ? and P = i^ r, 



§206.] ELASTICITY AND STRENGTH OF EXTENSION. ETC. 383 

we obtain the worh to he dpne in stretching it to the limit of ekuh 
ticity a 



-X- 




in which V denotes the volume Fl of the body and A a number, 
given by experiment, which is called modulus of resilience for 
extension and is determined by the expression 

A = iA C. CD=ia T=^/^' 



E 



1(7'^. 



In like manner the worh necessary to compress it to the limit 
of elasticity is 

A = VA,, 
in which 

denotes the modulus of resilience for compression at the limit of 

elasticity. 

Similar formulas can be employed for the work done in tearing 

or crushing prismatical bodies ; for the first case we have 

L = VB, 
and for the second, 

B = the surface A UW denoting the modulus of fragility for 
tearing ; and B^ — the surface A Ux Wi, the modulus of fragility 
for crushing. 



3^4 GENERAL PRINCIPLES OF MECHANICS. [§207. 

We see from the foregoing that the mechanical effect necessary 
to stretch or compress a prismatical body to the limit of elasticity, 
as well as that, which is necessary to produce a tearing or crushing 
of the same, is not at all dependent upon the different dimensions, 
but only upon the volume Fof the body; that, EoG., for two prisms 
of the same material the expenditure of mechanical effect in pro- 
ducing rupture is the same, when one is twice as long as the other 
and the cro€S-section of the former but one-half that of the latter. 

Example.— if the modulus of elasticity of wrought iron is ^ = 28000000 

pounds and the extension of the same at the limit of elasticity a = t^t^, 

T 

the modulus of proof strength is, smce o- = — , 

28000000 , . . 

T = a E = — zr^T^ — = 18700, (approximatively) 

and consequently the modulus of resilience for extension is 

T"" 18700 

^■■■=^<'T=^^ = ^o'^ E= g^-j-g-QQ = 6,23 pounds. 

Hence, in order to stretch a prismatical body of wrought iron to the limit 
of elasticity, the mechanical effect 

L ^= AV = 6,23 Fis necessary. 
If, E.G., the volume of this body were F = 20 cubic inches, the me- 
chanical effect would be L — 6,23 . 20 = 124,6 inch-pounds = ' 
= 10,38 foot-pounds. 

(§ 207.) Extension of a Body by its Own Weight. — 

If a prismatical body A B, Fig. 315, has a considerable length I, 
it undergoes, in consequence of its weight, a notable extension, 
which can be determined in the following manner. Let F denote 
the cross-section of the body, y its heaviness or the weight of a cu- 
bic inch of the matter composing it and x the variable length of a 
portion of it ; the tension in an element M N\^ produced 
Fig. 315. y^-^ ^^^ weight of the part of the body B M lying below 
A it, and consequently [according to § 204, (2)] the cor- 
responding extension of the length M N = ^ x oi ihh 

"Sm element is 

-ivitii J-, 

7 2 _ y F X n __ y -J 

Cv A/ rrr == Ct X __ X CL Xt 

F E E 

By integration we obtain the extension of the entire 

^ piece B M 

. y r n y x^ 
X = ~- I X dx — 
E t/ 



^e/ 2 E' 

and consequently that of the entire body ^ ^ is 



§207.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 385 

_ yV _ y FV ^^,G 
2^ ~ ^FE FE ' 

in which G = y Fl denotes the weight of the whole body. 

If this weight was not equally distributed in the body, but 
applied at its end B, the extension would be 

^' "" Te ^^^' 

The extension /I = ^ A^ of a body in consequence of its own 
weight, is but one half as great as that produced by the same weight 
at the end of the body. 

The same law obtains of course for the compression X produced 
in a body by its own weight. 

If in either case a pull or thrust P acts upon the body, we have 
the extension or compression produced 

PJ_ ^l_ _ {P ±i G)l 

' FE '' FE FE 

in which the upper sign is to be employed, when the force P acts 
in the same direction as the weight G, and the lower one, when it 
acts in the opposite direction. In the latter case, the extension is 
of course smaller than when P is the only tensile or compressive 
force. , 

The total extension or compression is = 0, when 

^a = P,or G = y Fl=2P,or 
2P 



1 = 



y F' 



The force P, acting at the end of the body, extends it equally 

A P 
in all parts, viz., in the ratio -j = ^r^, while, on the contrary, the 

weight G stretches or compresses it in the variable ratio -,-— — ^— . 

dx E 

The ratio of the total extension at any point, at the distance x from 

the point of application of the force P, is 



Aj A, CL A, 

I ~ I dx 



= (f =^^^)¥- 



If the force P acts in the same direction as G, the maximum 
ratio of extension or compression is for x — I, and it is then 

ii _ /P , ,\ 1 _ P + G 
I \F ^ '^ 1 E ~ FE 
25 



386 GENERAL PRINCIPLES OF MECHANICS [§307. 

and, on the contrary, the minimum is for x = 0, 1.E., at the point 

X P 

of application of P, and it is -^ = ^rrf 

K P and G act in opposite directions, we must distinguish the 

P . . P 

cases, in which I < -^ — and in which I > -j^ — . In the first case 

the ratio of extension or compression -y =z i— — y x\ ^ \^ a 

P 

maximum for a; = and = ^ ^ , and a minimum and = 

iP \ 1 

/ — — y Z I — for ic = Z. In the latter case there is a positive 



maximum ^r^ for x = 0, and a negative maximum lyl — -^\^ 

p 

for X = I, and, on the contrary, for x = -= — the function becomes 

= zero. 

In order that the body shall be extended or compressed to the 

limit of elasticity only, the maximum of the ratio of extension or 

/P \ 1 T 

compression ( -=- ± y a; I -=- should be at most = g = — , or more 

simply the maximum of (^ =t: y a;) -— 7^. But, when P and G 

have the same direction, this maximum is 

_ P P + yFl _ P + G 

-p-^y^- F ~ ~~F~'' 

p -\- y Fl 
and therefore we must put ^ = T, or P = F {T — y I), 

hence the required cross-section is 

F= ^ 



T- yX 

If, on the contrary, the forces P and G act in opposite directions, 

p I P\ 
we have two maxima, one = -^ and the other = \y I =-), and 

therefore the corresponding cross-section is equal to the greater of 

the values ^ P ^ -^ P ^ 

F= — and F = —^ — T. 

T yl 

If in the formulas we substitute K instead of T, we obtain the 
conditions of tearing and crushing, that is, in the first case, 
P = F (K — y /), and in the second either 
P = FKoY P = F{yl-K). 



N 



§208.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 387 

For P = we have either 

yl- T= andZ = -or 

y 

yl — K— and l——\, 

y 

the first formula being applicable to the case, when the body is ex- 
tended or compressed to the limits of elasticity, and the second to 
the case, when a tearing or crushing of the body takes place. 

Remark. — The energy stored by a body, which is extended or com- 
pressed by its own weight, can be calculated in the following manner. The 
element M N, Fig. 316, whose length is dx^ is gradually stretched by the 
weight y F X oi the portion of the body 5 if an amount, which 

Fig. 316. . ^ y xdx 

increases gradually from to <* /i = — — — , and toe work done 

in accomplishing it is 

Mi "" \l Fx. 61 = \—^- ^^^ 

Integrating this expression, we ootain the expression for the 
quantity of work done in extending all the elements of the rod 
from B to if, 

and that done in extending the entire rod 

y^Fl"^ y^F^Pl 0^1 

^~^' dF ~^' SFF ~^'dFB~^ ' 

C J 
in which (according to § 207) ^ = ^ jji et denotes the total extension of 

the rod. 

Example. — If a lead wire, whose modulus of rupture is ^ = 3100 and 
the weight of a cubic inch of which is = 0,412 pounds, is suspended verti- 
cally, it will break by its own weight, when its length is 

TT ^1 00 
l=~ = --— = 7524 inches = 627 feet. 
7 0,412 

If the modulus of proof strength is T = 670, it is stretched to the limit 

of elasticity, when its length is 

T 670 
Z, = — = ,— — = 1626 inches = 135,5 feet, 

^ y 0,412 ' 

and if its modulus of elasticity is ^ = 1000000 pounds, we have for the 

corresponding extension 

T 670 

l = —l^= . 135,5 = 0,090785 feet = 1,0894 inches. 

§ 208. Bodies of Uniform Strength. — If the pull or thrust 
P upon a vertical prismatical body is sensibly augmented by its 
weight G, we must of course put 

P+ G = FTorP=^FT- G= F(T~ly), 



388 GENERAL PRINCIPLES OF MECHANICS. [§208. 

and determine the cross-section of this body by means of the for- 
mula (compare § 207) 

P 

T-ly 

If this body, as, E.G., A B^ Fig. 317, is composed of prismatical 
parts, we can save material by giving to each of these parts a cross- 
section calculated by means of this formula. If the 
length of these portions of the body are Zi, 4, 4, etc., and 
if the load P is gradually increased by the weights F^ ^ y, 
F<i I2 y, F^ Z3 y, etc., of the portions to P^, P^, P3, etc., the 
required cross-section of the first portion is 

F. = -^—' 

that of the second should be 

P, F,T . 



bT F, 



T~ l,y T~l,y' 
that of the third 

^' = yiri^ = r:r77y etc. 

If the length of all the parts is the same, or /j = 4 = l^, etc., = I, 
we have more simply 



' f -ly ' T \T- lyl 
_ Pi T _ P T _ P / T V 

' ~ T - ly ~~ {T- lyf~ T \T -lyJ ' 

F,T P ( T \' ^ 

-^3 = Tfr^-T-7. = ^ iTn ^ , etc., 



or in general for the cross-section of the ^th portion 
P / T 

lyj 

If the cross-section of all the pieces are to be the same, that 
cross-section should be 



" T \T -lyl' 



T-nly T \T-nlyl 



T-nly T \T-nly) 
While in this case the volume of the whole body would be 

1 — nly 
in the former case, where every piece has its own proper cross-sec- 
tion, the volume is determined by the geometrical series 



§208.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 389 

Vn - {F, H- F, + .-. . + i^J / 

pj _ - _ ^ /_ y y , , / y y-n 

~ T- ly l ^ f- I y "^ \T- I yl ''' " ' '^ \T - I yl S 

But the sum of the geometrical series in the parenthesis is (see 
Ingenieur, page 82) 

whence it follows, that 

Pa ^ V i1_ (^n- F,) T 



-■ - ~ [(r^.) - ■] 



y 

and that the weight of the whole body is 

G = (i^„ - F,) T. 

If , the length I of the parts is very small, and, on the contrary, 
their number 7i yery great, and if we denote the total length n Ihj 
a, we have, reasoning as in § 194, 

in which e = 2,71828 is the base of the Naperian system of loga- 
rithms, and therefore we have 

P I T Y_ P_P«_v_^«v 

1 e rj{ 



in which F^ — -j^ denotes the area of the first cross-section. 



P 
T 

We have also approximatively 

P \^ ay 



and, on the contrary. 



F = 



F=t' 



a 

1 + -Tff 



TV ' T ^ \T ) X 
The volume of the body, composed of very many small por- 
tions, is found in the manner shown above to be 

■'.=f[(^f)--i]=f(--o. 

approximatively 



390 



GENERAL PRINCIPLES OF MECHANICS. 



[§208. 



while on the contrary, the volume of the bo&y with a constant 
cross-section is approximafcively 

Pa _^r-, ,^, /^Vl 



V 



T 

The formulas 



ay 



F = 

-*■ n 



P ^ 

-^e^ and F„ 



ay 



Fig. 318. 

A 



hold good, of course, for every body, such as A B, Fig. 318, and 
A B, Fig. 319, in which there is a constant variation of the cross- 
section. In order to find the cross-section F^ 
for any position M and the volume of the body 
cut off at the same point, we have only to sub- 
stitute in this formula for a the distance B M 
of the given position from the point of applica- 
tion B of the tensile or compressive force. The 
bodies thus determined have at every point a 
cross-section corresponding to the load they 
support, and are therefore called todies of uni- 
form strength (Fr. solides d'egale resistance, 
Ger. Korpervon gleichem Widerstande). These 
bodies have (the other circumstances being the 
same) the smallest volume, require therefore the least quantity of 
material and are for this reason generally the cheapest and most 
advantageous that we can employ. If we compare such a body 
with a prismatical one, we find from the above approximate formu- 
las, that the economy of volume is 

"~ T 12 T ^ 6\T/ ]~ 2T' \ '^ S T J' 

Remark, — Since the relative extension and compression of a body of 

T 
uniform strength is everywhere the same, viz., cr = -=,, its total extension is 

T 

7^ = a a = --=,a^ while for a prismatical body it is only 




A = 



{P + ^ G) a P + i G T 



^^• 



FE ~ p + a 

Example. — What must be the cross-section of a wrought-iron pump 
rod, whose length is 1000 feet, when, in addition to its own weight, it must 
support a load P = 75000 pounds ? If instead of the modulus of proof 

T 

strength T= 18600 we employ for safety a working load -^r = 9300 pounds 



3 



and put the weight of a cubic inch of wrought-iron 

7,70 . 62,425 



§ 209.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 391 

the required cross-section is 

P 75000 75000 ,^ ,^ . , 

^ = 2^31^ = 9300 - 12 00 . 0,2782 = 5962 = ^^'^^ '^"^'" ^^^^^^' 

and the weight of the rod is 

G = F .ay = 12,58 . 12000 . 0,2782 = 42000 pounds. 

If we could give this rod the form of a body of uniform strength, we 
would have for the smallest cross-section 

P 75000 ^^, 
^0 = Y ^ "9300 ^ ' ®^^^^^ inches, 

and for the greatest 

F^ = 8,06 . 6^782. i.-jg ^ g^06 e'''"^' = 8,06 . 1,432 == 11,54 square inches, 
and the weight of the rod would be 

G„ = V„y= {Fn- F) T = (11,54 — 8,06) 9300 = 32364 pounds. 

If the modulus of elasticity of wrought iron is ^ = 28000000 pounds, 
the extension of the rod in the latter case would be 

^ T 18600 . 1000 186 93 ^ ^ « n<^ • -u 

^ = ^ ^ = -^00000- = 280 = 140 ^^^* = ^'^^ '°"^''' 
and, on the contrary, in the first case it is 

P + i ff , 75000 + 21000 , _ 96000 _ „„ „ ^, . , 
-TVa ^ = 7^5000-Tl3000 • ^'»^ = imOO • ^'^^ = 6-54 inches. 

§ 209. Experiments upon Extension and Compression. 

■ — In order to study thoroughly the laws of the elasticity of any 
substance, it is necessary not only to submit prismatical bodies of 
this substance (which should be made as long as possible) to 
extension or compression by weights, which are gradually increased 
in amount until rupture is produced, but also to observe the exact 
extension or compression produced by each weight. If we place 
the bodies to be experimented upon in a vertical position, the 
Aveights can be hung or laid upon them, and they then give 
directly the pull or thrust to which the body is subjected. In 
order to avoid experimenting with too great weights, we generally 
prefer to let the weights act upon the body by means of a lever 
with unequal arms ; the weights are always hung upon the long 
arm (a), and the body is acted upon by the shorter arm (J). Mul- 
tiplying the weight G by the ratio -7 of the arms, we find the corre- 
sponding pull or thrust P =. :-Q. The so-called hydraulic press 



392 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 209. 



can also be employed with advantage instead of weights to produce 
very great tensile or compressive forces. In order to observe the 
amount of the extension or compression, a fine line is drawn upon 
the bar to be experimented with near each of its ends, or a pair of 
pointers, with verniers attached, are fastened to it at those points, 
and in order to determine not only the elastic, but also the perma- 
nent extension or set, we measure the distance between these lines 
or pointers not only before and during the application of the 
weights, but also after they have been removed, and it is generally 
preferable to allow several minutes or even hours to elapse between 
the application or removal of the weights and the measurement ; 
for when the forces are very great the extension and compression 
do not assume the true value in a moment, but only after a certain 
time. This distance is measured either with a bar compass or 
directly by means of a division on the rod itself The so-called 
cathometer is also employed for this purpose ; it consists essentially 
of a vertical staff and of a spirit-level, which is capable of sliding 
up and down the former (see Ingenieur, page 234). In order to 
observe the compression on long rods, we must enclose them in 
tube-shaped guides ; they must also be well greased from time to 
time, so that they can slide without resistance in their guides. 

If we wish to determine the modulus of ultimate strength of a 
body, we can employ shorter pieces for the experiments. In 

experimenting upon ruptnre hy 
extensiony^Q employ bodies with 
large heads A and B, Fig. 320, 
through which holes are bored 
exactly in the axis. In the 
middle of each hole a circular 
knife-edge is made, so that the body shall be pulled exactly in the 

line of the axis by means of the bolt CD and the 
clevis F E, which is applied to its ends. 

In experimenting upon rupture hy crushing, 
the two bases of the body {A, Fig. 321) are 
made parallel, it is. then brought between two 
cylinders B and (7, whose bases are ground flat ; 
while the rounded head of one of the cylinders 
is acted on by the compressive force, the other , 
is supported by the large bed-plate Z>, and both 
slide in the interior of cylinder E F. The 
pressure P upon the head H of the cylinder is 



Fig. 320. 




Fig. 321. 




§ 210.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 



393 



produced either by a hydraulic press or by a one-armed lever 
L 0, such as is partially represented in the figure. 

While the rupture of a body by tearing occurs in the smallest 
cross-section, and the body is therefore divided in two parts only, 
the rupture by crushing takes place generally in inclined surfaces, 
and the body is divided into several pieces. Prismatical bodies are 
divided, in the first place, into two pyramids, whose bases are those 
of the body and whose apexes are at its centre, and in the second 
place, into other pyramidical bodies, whose bases form the sides of 
the body and whose apexes are also situated at its centre. Bodies, 
whose structure in different directions is different, of course do not 
act thus ; e.g., a piece of wood would be compressed by a force 
acting in the direction of the fibres, in such manner, that at its 
smallest cross-section the fibres would be bent out in a spherical 
form. 

§ 210. Experiments upon Extension. — We are indebted 
to Gerstner for the first thorough experiments iipon the extension 
and elasticity of iron wire. He employed in his experiments iron 
wire from 0,2 to 0,8 lines in diameter and made use of the lever 
apparatus represented in Fig. 322 with the pointer CD 15 feet 

Fig. 322. 




long, the counter-balance G and the sliding weight Q. The wire 
E F, which was about 4 feet long, was firmly fastened at one end £J 
iind the other was wound round a pin F, which was turned by the 



394 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 310. 




endless screw /Si so that the wire could be subjected to any desired 
strain. The extension of the wire was shown by the pointer D 
■upon a rod ^ ^ in 54 times its natural size. The knife-edge C of 
the lever, the pin F, around which the upper end of the wire is 
wound, and the endless screw S, which turns the pin, are all repre- 
sented on. a larger scale in Fig. 323. 

Gerstner proves by his ex- 
periments, that every extension 
is the sum of two extensions, one 
of which {the elastic extension) 
disappears, when the weight is 
removed, and the other (the per- 
manent extension, or set) remains, 
so that the extension X is not ex- 
actly proportional to P within the limits of elasticity, and that it 
is more proper to replace the formula 

P = jFE [§204(4)] 
by the following series 

in which a and /3 are numbers determined by experiment. 

Quite extensive experiments upon the elasticity and strength 
of wrought iron and iron wire were afterwards made by Lagerhjelm 
and by Brix. Both experimenters employed in their researches a 
bent lever A C B, Fig. 324, the longer arm B of which was de- 
pressed by the weights G, which were laid upon a scale-pan W, and 




-^-j^=- 



§ 210.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 395 

thus the iron bar or wire D E, which was fastened to the shorter 

arm C A, was stretched to any desired extent. In the apparatus 

C A 
. used by Brix, the ratio of the arms of the lever was j^-^ = -j^, 

and one end D of the wire was attached to the arm C A with 
clamps, hooks and bolts, and the other end was fastened in the 
same way to a screw S, which was turned by means of a train of 
wheels by a crank K. The increase in length was given by two 
verniers, which were screwed fast to the ends of the wire and 
moved along two scales divided into quarter lines. When the wire 
had been firmly fastened in the clamps, the scale-pan was gradually 
loaded with heavy weights, and in each experiment the wire was 
stretched by turning {h.Q crank K until the lever was lifted from 
its support and the tension of the wire balanced the weight G. 
The experiments were made with wire 1^ to 1^ lines thick and 
gave for the average value of the modulus of rupture of unannealed 
wire K = 98000 pounds, and, on the contrary, after annealing, 
K = 64500 pounds. The average modulus of elasticity, on the 
contrary, for annealed and unannealed wire was found to be 
E = 29000000 pounds ; it was also found, that the limit of elas- 
ticity was reached, when the strain was 0,5 K for unannealed and 
0,6 K for annealed wire. 

When the tensions were greater, the extension became perma- 
nent, and the total extension of unannealed wire at the instant of 
rupture was 

X .A 

- = 0,0034, and that of annealed wire j = 0,0885, 

or 26 times as much. In the apparatus used by Lagerhjelm the 
tension on the wire was produced by a hydraulic press, the piston 
rod of which was attached to the end of the iron bar. 

Lagerhjelm employed in his experiments iron rods 36 inches 
long, ^ inch thick, the cross-sections of which were circular and 
square. According to his experiments, the average modulus of 
elasticity for Swedish wrought iron is 

E = 46000000 pounds ; 
the modulus of rupture or of ultimate strength is 

K= -~-E= 92000 pounds ; 
oUU 

and the modulus of proof strength 

T =a .E= -^ . 46000000 = 28750 pounds. 
IbOO 



396 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 210. 



Wertheim, in his experiments upon the elasticity and cohesion 
of the metals, allowed the wire to hang freely, and fastened to the 
end of the same a weight-box, which was supported upon the floor 
by means of feet, which could be raised or lowered by turning a 
screw. In order to stretch the wire by means of the weights 
placed in the box, the foot-screws were turned until the box swung 
freely. A cathometer was employed to determine the extension of 
the wire. 

The experiments were performed at very different tempera- 
tures, and with wire made of various metals, such as iron, steel, 
brass, tin, lead, zinc, silver, etc. The principal results of these ex- 
periments will be found in the table given in § 212. 

The apparatus, with which Fairbairn performed his experiments, 
consists essentially of a strong wrought-iron lever or balance-beam 
A C D, Fig. 325, whose fulcrum D is firmly retained by a strong 
bolt F, which can be raised or lowered by means of a nut. Two 

Fig. 335. 




iron pillars give the necessary resistance to the bed-plate IT H, 
througli which impasses. The piece of iron L M to be experi- 
mentecL upon is suspended by means of a chain to the support K K, 
which reposes upon the two columns 2' T and is connected by a 
bolt and clevis to the stirrup C of the lever A C D. To the longer 



211] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 397 



arm of the latter there is suspended not only a constant weight G^ 
but also a scale-board for the reception of smaller weights ; the 
bolt X serves to support the lever, and the latter is raised by means 
of a rope P, which passes over a pulley and is wound upon the 
shaft W of a windlass U Y Z. After the weights had been laid on, 
the arm E of the lever was allowed to sink gradually by turning 
the crank U, until the piece of iron to be tested w^as finally sub- 
jected to the tension produced by iV^ and G, 

Remaek. — Gerstner's experiments upon the elasticity of iron wire, etc., 
are discussed in Gerstner's Mechanics, Vol. I. For the experiments of 
Lagerhjelm, see Pfaff's translation of the treatise : Researches for the pur- 
pose of determining the density, homogeneity, elasticity, malleability, and 
strength of bar iron, etc., by Lagerhjelm (Niirnberg, 1829), and tlie informa- 
tion in regard to the experiments of Brix is to be found in the treatise on 
the cohesion and elasticity of some of the iron wires employed in the con- 
struction of suspension bridges (Berlin, 1837). 

The experiments of Wertheim upon the elasticity and cohesion of the 
metals, etc., as well as of glass and wood, are discussed in "Poggendorf 's 
Annalen der Physik und Chemie," Erganzungsband II, 1845. In the 
latter experiments the modulus of elasticity of the bodies named was de- 
termined not only by experiments upon extension, but also by experiments 
upon flexion and vibration. For Fairbairn's experiments on the strength 
of materials, his ''Useful Information for Engineers" can be consulted. 

§ 211. Iron and Wood. — The most complete set of experi- 
ments upon the elasticity and strength of cast and wrought iron 
are those more recently made by Hodgkinson. By these we have 
for the first time acquired a complete knowledge of the laws of ex- 
tension and compression for these materials, which are of such 
great importance in their practical applications. Although, accord- 
ing to these experiments, iron produced in different ways has 
difierent degrees of elasticity and strength, yet it is possible to 
express the behavior of this body in regard to extension and com- 
pression by means of curves. 

The average modulus of elasticity of cast iron (Fr. fonte, Ger. 
Gusseisen) is, according to these experiments, for extension as well 
as for compression 

E = 1000000 kilograms, when the cross-section is one centime- 
ter, and consequently 
E = 14,22 . 1000000 = 14220000 pounds when the cross-section is 

one inch. 

The extension at the limit of elasticity is 

_ A _ _1_ 

"^ ~ I ~ 1500* 



398 GENERAL PRINCIPLES OF MECHANICS. [§ 211. 

This extension corresponds to the modulus of proof strength 
T — -T^^~ = 667 kilograms, or 

r=y^ = 9480 pound. 

The compression at the hmit of elasticity, on the contrary, is 

_ J. 

"^^ "~ 750' 

and therefore the modulus of proof strength is 

^ 1000000 -.oQQi-i 14220000 ,_^^. , 

Ti = — =^^r — = 1383 kilograms = — — r — = 18960 pounds. 

The modulus of rupture for tearing was found by these experi- 
ments to be 

K = 1300 kilograms = 18486 pounds, 

and, on the contrary, that for crushing 

Ki = 7200 kilograms = 102400 pounds. 

The resistance of cast iron to crushing is, therefore, 5J times as 

great as that to tearing. 

For wrought iron (Fr. fer; Ger. Schmiedeisen) we haye for 

extension as well as compression 

JEJ = 2000000 kilograms = 28440000 pounds, 

/I 1 

and the limit of elasticity is reached, when a = j = ^¥7^? whence 

the modulus of proof strength is 

^ 2000000 ,000 Ti -,QQ«A A 

T = - = 1383 kilograms = 18960 pounds. 

loUU 

Finally the modulus of rupture or of ultimate strength of 
wrought iron was found to be for tearing 

X = 4000 kilograms = 56880 pounds, 
and for crushing 

Ki = 3000 kilograms = 42660 pounds. 
The modulus of elasticity of wrought iron is therefore about 
double that of cast iron, and while the modulus of rupture by tearing 
of cast iron is but about ^ that of wrought iron, the modulus of rup- 
ture by crushing of cast iron is nearly 2^ times as great as that of 
wrought iron. The relations of the elasticity and strength of cast 
and wrought iron are graphically represented in Fig. 326. From 

the origin A on the right-hand side of the axis of abscissas X X 
the tensile forces, given in thousand pounds per square inch, are 
laid off and on the left-hand side the compressive forces, while the 



§ 211.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 39^ 



upper half of the axis of ordinates Y Y represents the correspond- 
ing extensions, and the lower half the compressions. It will at 
once strike the eye, that the curve of cast iron has a great develop- 
ment on the side of compression and that of wrought iron on the 
side of extension ; and we also remark, that the curves form 
approximativelj straight lines near the origin A. 

Fig. 326. 



Thousandths IQlY 
9- 



Wrought Iron 



Thousand pounds 
30 





10 20 30 40 
Thousand pounds 



60 



Cast Iron 

_Y-^I0 ThousandthJi 

As next to iron wood (Fr. bois ; Ger. Holz) is most generally 
employed in construction, the relations of the elasticity of fir, 
beach and oak wood are graphically represented in the figure by 
a curve. The average modulus of elasticity of these kinds of 
wood is 

E = 110000 kilograms = 1564200 pounds. 

The limit of elasticity is reached, when a =—- of the length, and 

600 

the corresponding modulus of proof strength is 

110000 ,^^,., 
gQQ =180 kilograms = 2607 pounds. " 



T = 



Finally, the modulus of rupture for tearing is 
E = 650 kilograms = 9243 pounds, 



400 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 211. 



and, on the contrary, for crushing 

K — 450 kilograms = 6399 pounds. 
The ratio 156 : 1422 : 2844 approximatiyely = 1 : 9 : 19 of the 
moduli of elasticity of wood, cast and wrought iron to each other 
is expressed in the figure by -the sub tangents ab, ac and ad. 

Fig. 327. 

Wrought Iron 
Thousandths 



Thousand pounds 
90 &0 70 60 50 40 30 20 10 





X 



Wrought Iron 



/A 10 20 30 40 
y Thousand" poiihas 

-2 



50 60 



Cast Iron ig 

_YMO Thousandths 

The modulus of resilience ^ = | tr T for the limit o\ elasticity 
is expressed by the triangles A a h, A a^ Ci and A a^ di, the bases 

of which are the small ratios of extension g =z A a = 7^7777 and 

oOO 



a — A a^ = 



( approximati vely). 



1500 
From the above, we have for wood 

A = ^a T = I , ^^^ . 180 = 0,15 kilogram centimeters 



for cast iron 



600 
= J . -x^ . 2607 = 2,17 inch-pounds, 



. 667 = 0,222 kilogram centimeters = 3,16 inch- 



- 1500 
pounds, and for wrought iron 



§ 212.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 401 

1333 

A = ^ . rr^T: =■ 0,444 kilogram centimeters = 6,32 inch-pounds. 

Properly, a complete series of experiments is necessary to deter- 
mine the modulus of fragility for tearing or crushing; for this 
modulus is found by the quadrature (see Art. 29, Introduction to 
the Calculus) of the complete branches of the curve on either side, 
and this is especially necessary for the extension of wrought iron and 
for the compression of cast iron, since the curves corresponding to 
the changes in these bodies differ considerably from right lines. 
The extension and compression of wood at the instant of rupture 
by tearing or crushing is so little known, that we are unable to 
give with any degree of certainty its moduli of fragility. If we 
treat the corresponding curve as a right line, we obtain the modu- 
lus of resilience for tearing 

B = ^ ^ = Tjj^T^ = lj91 kilogram centimeters = 27,2 inch- 

pounds, and, on the contrary, the modulus of fragility for crushing is 

77' i 450^ 

B=}^ -^=-1 . = 0,92 kilogram centimetres = 13,07 inch-lbs. 

When cast iron is ruptured by tearing, assuming the extension 
to be (Ji = 0,0016 and the mean value of the force to be 560 kilo- 
grams, the modulus of fragility is 
B — 0,0016 . 650 = 1,04 kilogram-centimetres = 14,8 inch4bs. 

When cast iron is ruptured by crushing, the maximum exten- 
sion can be assumed to be a^ = 0,008 and the mean crushing force 
to be = 3600 kilograms; hence the corresponding modulus of 
fragility is 
B, = 0,008 . 3600 = 29 kilogram-centimetres = 411 inch-lbs. 

We can assume as the mean value of (7^ for the rupture of 
wrought iron by tearing, 0,008 and for the mean value of the 
force 3000 kilograms; hence the corresponding modulus of fra- 
gility is 

B = 0,008 . 3000 = 24 kilogram-centimetres = 341 inch-lbs. 

On the contrary, for the rupture of wrought iron by crushing, 
we must assume a = 0,0018 and the mean force to be = 1300 
kilograms; whence the corresponding modulus of fragility is 
B = 0,0018 . 1300 = 2,34 kilogi-am-centimetres = 33,3 inch-lbs. 

§ 212. Numbers Determined by Experiment. — In the 

following tables I and II the mean values of the moduli of elas- 

20 



402 GENERAL PRINCIPLES OF MECHANICS. [§213. 

ticity, of proof strengtli and of ultimate strength of the materials 

generally employed in constructions are given. The first table is 

for tensile and the second for compressive forces. 

X 
The value of the relative extension cr = - for the limit of elas- 

V 

ticity given in the second column of the tables expresses also the 

T 

ratio ^ of the values of T and E given in the third and fourth 

columns. In practice the bodies are only loaded with ■ — T, e.g., 

\ Tto \ T, on the cross-section is determined by substituting in 
the formula 

instead of K, for metals the modulus of safe load - K =^ ^ X, for 

lb 

wood and stone = jq K^ and for masonry but -^-^ K. On the con- 
trary, for ropes we can employ ^ ^ to i ^. We call n a factor of 
safety. 

The lower numbers in the parenthesis -j >- give the values in 

kilograms, assuming a cross-section of 1 centimetre square; the 
upper numbers express the values in pounds referred to a cross- 
section of one square inch. 

Remakk. — The moduli given in these tables are for unannealed metals. 
For annealed metals (Fr. metaux cuits, Ger. ausgegliite Metalle) the modu- 
lus of elasticity is generally the same as for unannealed metals, while the 
modulus of rupture by tearing of annealed metals is generally from 30 to 
40 per cent, less than that of unannealed ones. Tempered and annealed steel 
(Fr. acier trempe et recuit, Ger. geharteter und angelassener Stahl) has the 
same modulus of elasticity as untempered steel, but its modulus of proof 
strength is 20 to 30 per cent, greater than that of untempered steel. When 
it is not otherwise stated, the moduli for metals were determined with 
wire, which had on the outside a harder crust (caused by the drawing) 
than hammered or cast metal rods. For some materials, e.g. wood, iron, 
and stone, the moduli of elasticity, of proof strength and of ultimate 
strength vary so much that in particular cases a value differing 25 per cent, 
(more or less) from those here given may be found. 



%2VZ.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 



403 



TABLE I. 



MODULI OF ELASTICITY AND STRENGTH FOR EXTENSION. 



Name of the material. 



Cast iron 

Wro't iron in rods. 



in wire 



in sheets. 



German steel, tern 
pered and annealed 

Fine cast steel. . . 
Hammered copper 
Sheet copper .... 

Copper wire 

Zinc, melted 

Brass 

Brass wire 

Bronze, gun metal. 

Lead 

Lead wire 



Extension 
\ 

"J. 

at the limit of 

Elasticity. 



1500 

I 



1500 

I 



1000 
I 



1250 

I 



I 
I 



4000 

I 



3650 

I 



1000 

I 



4150 
I 



1320 
I 

742 
I 



1590 
I 

477 

I 



1500 



0,000667 

0,000667 

0,001000 

0,000800 

0,001198 

0,002222 

0,000250 

0,000274 

0,001000 

0,000241 

0,000758 

0,001350 

0,000629 

0,00210 

0,000667 



Modulus 
of Elasticity B. 



14 220000 
I 000000 

28 000000 

1 970000 

31 000000 

2 190000 

26 000000 

1 830000 

29 000000 

2 050000 

41 500000 
2 920000 

15 640000 
I lOOOOO 

15 640000 
I lOOOOO 

I 720000 
I 210000 

13 500000 
950000 

9 lOOOOO 
640000 

14 000000 
987000 

9 800000 

690000 

7IIOOO 
50000 

I 000000 
70000 






9480 
667 

18700 

1313 

31000 
2190 

20800 

1475 

34730 
2460 

92200 
6490 

3910 

275 

4285 

301 

1720 
1210 

3250 
229 

6890 

485 

18900 

6160 

434 

1490 

105 

667 

47 



Modulus of 

Resilience 

A ^ i<T T. 


3.16 


0,222 


6,23 


0,44 


15.5 


1,10 


8,32 


1,18 



20,8 

1,48 

102,4 

7,20 

0,49 

0,034 

0,59 

0,041 

8,60 
0,605 

0,392 

0,029 

2,61 
0,184 

12,76 
0,90 

1,94 
0,136 

1,56 
0,110 

0,22 
0,016 



i2 rt r; 



18500 
1300 

58200) 

4090 X 

88300 

6210 
46800 

3290 

II6500) 
8190) 

145500 1 

10230 j 

33800]^ 

2380] 

30400} 

2140 J 
60300 ) 

4240 j 

7500 
526 

17700 
1242 

51960) 
3654) 

36400 
2560 

1850 

130 
3100 

220 



40t 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 313. 



MODULI OF ELASTICITY AND STRENGTH FOR EXTENSION— (7o?i^w^^ed 



Name of the material. 



Tin 

Silver 

Gold 

Platina 

Aluminum 

Glass 

Wood : beach, oak, 
pine, spruce, fir, 
in the direction 
of the fibres .... 

The same kinds of 
wood in the di- 
rection of the 
radii to the 
yearly rings 

The same kinds of 
wood parallel to 
the yearly rings . 

Light hemp rope . . 

Strong hemp rope . 

Wire rope 

Chain cable 



Leather straps (cow 
leather) 

Sheet iron (riveted 
I with one row of 
] rivets) 



Extension 

A. 

a — — 

I 

at the Umit of Elasticity. 



900 
I 



660 

I 

600 

I 



600 



0,001111 
0,001515 
0,001667 
0,001667 



600 



= 0,001667 



Modulus 
of Elasticity E. 



5 700000 

400000 

10 400000 

730000 

11 400000 
800000 

22 800000 
I 600000 

9 6000000 

675000 

10 000000 
700000 



I 560000 
IIOOOO 



1 ■; 



85000 
3000 



II40OO 
8000 



10400 



O r- 



^^ 



SO ^b 
'w o ^ 
^ ^- !" II 
^ ft-C " 
.9 t«fL, 



6300 

440 

15800 

IIOO 

19000 
1300 

38000 
2700 



2600 
180 



en p , 
;3 O) b 

-BJii 



3.50 
0,24 

12,00 
0,83 

1,09 

31,7 

2,25 



2,17 
0,15 



O !vi 
3 • n J-^ 



m 



5000 
350 

41200 
2900 

38400 

2700 

48300 
3400 

28900 
2030 

3530 1 

248; 



9200 

650 



570 
40 



640) 

45) 
8700 
610 

6830 
480 

47000 f 

51900) 
3650) 



.100 \ 

290 j 



37000 
2600 



1 212.] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 405 

TABLE 11. 
THE MODULI OF ELASTICITY AND STRENGTH FOR COMPRESSION. 



Name of the 
material. 


Compression 

A 

at the limit of elasticity. 


Modulus of elasticity K 


Modulus of proof strength 


4) 
U 

a 

§ II 




"So 
s 
2i 

V 

H 
S 

in 

1 


Cast iron . 


= 0,001333 
750 


f 14000000 
( 990000 


18700 
1320 


12,44 
0,88 


104000) 
73^0) 


Wrought " 


/-\ /-\i^/-\A^H 


( 28000000 
( 1970000 


18700 
1320 


6,23 
0,44 


31000) 
2200) 


U.UUUUU/ 

1500 


Copper . . 


I 

— r\r\r\^ T" r\ 


( 15640000 
( IIOOOOO 


3910 

275 


0,49 
0,039 


58300) 
4100 ) 


4000 


Brass . . . 










j 10400 I 

X 731 \ 


Lead .... 










{ To} 


Wood in 












the direc- 












tion of the 










J 6800 ) 

1 480 f 


fibre. . . . 










Basalt . . . 










j 28000 ) 

i 1970 \ 


Gneiss and 










i 83001^ 


granite . . 










1 585 1 


Limestone. 










\ 5200 I 
i 365 s 


Sandstone . 











j 4150 [ 
( 292 \ 


Brick . . . 










1 830 / 
' 59 \ 


Mortar. . . 










\ nti 



Example 1. What should be the cross -section of a wrought-iron rod 
1500 feet long, which is subjected to a pull of 60000 pounds ? 

T 

Neglecting the weight of the rod and allowing a strain of — = 9350 



ttf"' 



"^"' ' . , . , . , . ^ 60000 

pounds per square inch, we obtain the required cross-section F = -^q^tt 

= 6,42 square inches. Taking into account the weight of the rod, the 
weight of a cubic inch of iron being y = 0,280 pounds, we have 



406 



GENERAL PRINCIPLES OF MECHANICS. 



[§213 



i^ = 



60000 



60000 6000 , 
"5040 ~ '43r~ ' ®^^^^® inches. 



9350 — 1500 . 12 . 0,380 9350 

The weight of the rod is G = Fly = 5040 . 13,93 = 70157 pounds, and 
the extension of the same by the pull P = 60000 pounds and by the weight 
G = 70157 pounds is 

_{P+ ^ G)l _ 95078 . 18000 _ 143617 _ 
'^ ~ We "~ 13,93 . 38000000 ~" 33480 ~ '^^ inches. 

Example 3. How thick must the foundation walls of a building 60 feet 
long and 40 feet wide on the outside, and weighing 35000000 pounds, be 
made when we employ good cut pieces of gneiss ? If we make the thick- 
ness of the wall equal to x^ we can put the mean length of the wall = 60 
— X and the mean breadth = 40 — «, and therefore the mean periphery 
3 . (60 — ic + 40 — ic) = 300 — 4: x, and consequently the base of the 
whole masonry is (300 — 4, x) x square feet ~ 144 (200 — 4 cc) ;» = 576 
(50 — x) X square inches. The modulus of rupture of gneiss for crushing 
is 8300 pounds. If, therefore, we assume a coefficient of security of ^ or 
a factor of safety of 30 for the wall, we can put the allowable pressure 

upon a square inch = — 415 pounds ; hence we have 

20 

415 . 576 (50 -x)x= 35000000, 

whence 5{)x— x^ = 146,4, 

and finally the required thickness of the wall 



'^\ 



X = 



146,4 +X'' ^^^^ 8,57 „,^^ , 
' ^ . = 3,938 + ^ = 3,10 feet. 



50 ' ' 50 

213. Strength of Shearing.— The strength of shearing (Fr. 
resistance par glissement ou cisaillement, Ger. Schubfestigkeit or 
Widerstand des Abdriickens oder Abscheerens), which comes into 
play when the surface of separation coincides with the direction 
of the force, can be treated in the same manner as the strength of 
extension. We have here to consider the action of three parallel 
forces P, Q, and E, Fig. 328, when the points of application A and 
Q of two of the forces lie so near each other, that bending is not 
possible, and therefore a separation of the body in two parts takes 

Fig. 338. Fig. 329. 



iHll 




IfkM^ 



tp pi 

place between A and C in a surface D Dut right angles to the axis 
of the body. The strength of shearing, like that of tearing and 




§213] ELASTICITY AND STRENGTH OF EXTENSION, ETC. 407 



crushing, is proportional to the section of the body, or rather to the 
area F of the surface of separation, and in the case of wrought iron 
is approximately equal to that for tearing, so that the modulus of 
rupture A'' for tearing can also be employed as the modulus of rup- 
ture for shearing, and consequently we can put the force necessary 
to produce rupture by shearing, when the cross-section is F, 
P ^= F K. In general we have P = FK^, K^ denoting the ultimate 
strength of shearing per unit of surface determined by experiment. 

X 
The formula P = j F E ^^ a F F for tensile and compressive 

forces within the limit of elasticity can also be employed for the 

C A 
shearing force P, Fig. 329, but here a denotes the ratio t = 7T~n 

of the displacement C A to the distance C B of the directions A P 

and E F of the two forces from each other. 

The following Table III. contains the modulus of elasticity {C) 

and that of rupture or ultimate strength {K^) for all bodies, for 

which they are known at present, and they correspond to the 

formulas P =^ t F C and Pi^= F Kc^ for the elasticity and strength 

of shearing. 

TABLE III. 

MODULI OF THE ELASTICITY AND ULTIMATE STRENGTH OF 

SHEARING 



Names of the Bodies. 



Cast Iron 

Wrought Iron .... 
Fine Cast Steel . . . 

Copper 

Brass ....... 

Wood of deciduous Trees . 
Wood of evergreen Trees . 



Modulus of Elasticity C. 



2840000 
200000 

9000000 
630000 

14220000 
lOOOOOO 

6260000 I 
440000 J 

5260000] 
370000 ) 

569000 
40000 

616000 

43300 



Modulus of Ultimate 
Strength A'o • 



32300 
2270 

50000 
3500 

92400 \ 
6500 j 



683 
48 

2290 
161 



C is generally taken = I E and K.^ = K. 



408 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 213 



The most important application of the formula P = F K^ is 
to the determination of the thickness d of bolts and rivets, with 
which plates and other flat bodies are fastened together. There 

are two modes in which bodies 



Fm. 330. 



I'iG. 831. 






may be fastened together in this 
way ; either the plates A B and 
CD to be joined together are 
laid upon one another, as in Fig. 

330, and then fastened together 
by the bolts or rivets JV^JVand 
0, or, as is represented in Fig. 

331, the plates are butted to- 
gether and covered with splicing 
pieces D D and E E, and they 
are then fastened together by 
means of the^rivets N N and 0, 

which pass through both the plate and the splicing pieces. In the 
first method of joining the plates the tensile stress passes from one 
plate to the other through the intervention of a couple, which 
causes both of the plates to undergo in addition to the stretching 
also a bending, and consequently their safe or working load is 
diminished. The second method, where no such couple is called 
into action and where, consequently, no bending takes place, is for 
this reason to be preferred. Since the plates and splicing pieces, 
which are thus joined, press upon each other with no inconsidera- 
ble force, the strength of the joint is considerably augmented by the 
friction arising from this pressure. For greater safety we disregard 
this action in determining the thickness of the rivets. On the other 
hand, the working load of the plate is diminished by the holes 
made for the rivets or bolts, and we must therefore take care that 
it is not exceeded by the working load of the rivets. If d is the 
thickness of the rivets and v their number, in the case of the joint 
in two plates represented in Fig. 331, we have for the working 



load of the rivets 



P — V 



7Td' K, 



n 



Kow, if h is the width and s the thickness of the pieces to be joined 
and Vi the number of the rivets in one row, the cross-section of the 
plate submitted to the force P is 

{b — Vi d) Sf and therefore we have P = (b — v^ d) s — , 



F 



n 



K denoting the modulus of rupture of sheet iron ; equating these 
two values, we obtain 



§ 214.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 409 

— J — K^ — {h — Vi d) s K, or 
^Cb - Vrd)sK 

n d' k; 

When the holes in the plates are punched, the strength of 
shearing must be overcome, but in this case the surface is not 
plane, but cylindrical. If s is the thickness of the plate and d the 
diameter of the hole in it, we have the area of the surface of 
separation 

F =. TT d s, 
and consequently the force necessary to punch the hole is 

P = FK, = 7T ds K,. 
(Compare in the "Civil Ingenieur," Vol. I, 1854, the article "John 
Jones' experiments on the force necessary to punch sheet-iron," by 
C. Borneman). 

Example — 1) An iron rivet 1|- inch thick can resist with safety, if 've as- 
sume ^3 = 1 . 50000 = 8300 pounds, a force 

P = -J- ^3 = 4(2) • ^^^^ = 4 = ^^^^^ pounds, 

and the force necessary to punch the hole through the sheet-iron, which is 
^ inch thick, is 

3 1 
Pj =: TT <Z s . ^3 = TT - . - . 50000 = 37500 tt = 117810 pounds. 

2) If two pieces of sheet-iron are to be joined together by a row of 
rivets, and if we denote the thickness of the plate by s and its width for 
each rivet by 5, we have 

Q) — d)s := —T-i whence 

4 s \ 4 s/' 

E.G., for <f = f and s = ^ inch 

6 = I M + —^1 = 5 inches. 



CHAPTER II. 

ELASTICITY AND STRENGTH OF FLEXURE OR BENDING. 

§ 214. Flexure. — The most simple case of flexure is that of a 
body ABC, Fig. 332, acted upon by a force A P = P, whose di- 
rection is normal to its axis A B, while the body at the same time 
is retained at two points B and C. Let I and Ix be the distances 



410 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 314. 



C A and C B oi. the points of application A and B from the cen- 
tral fulcrum or point of application C, then the force at B is 

and consequently the resultant is 

i2 = P+g=(l + ^)p. 



Fig. 332. 



R 



j| fi|iiiiiimiiiiiiiiiiiiiiiiiiiiiiii iiii 



.^..1 



rs 



Fig. 338. 




If we wish to prevent one portion of the body from bending, 
we must insert between the two points of support an infinite num- 
ber of others, or the body must be fastened or solidly walled in 
along B C, as is represented in Fig. 333, and we have then to study 
only the flexure of the free portion A C of the body. Let us sup- 
pose the body to be a prism, and let us assume, that it is composed 
of long parallel fibres placed above and alongside of one another 
and that, when the body is bent, they neither lose their parallelism 
nor slide upon one another. 

By this flexure those fibres, which are on the convex side of 
the body, are extended, and those on the concave side are com- 
pressed, while a certain mean layer undergoes neither extension 
nor compression. This is called the neutral surface of a deflected 
'beam (Fr. couche des fibres invariables, Ger. neutrale Axenschicht). 
The extension and compression of the various fibres above and 
below this layer are proportional to their distance from it. The ex- 
tension of the fibres on one side and the comjoression of those on 
the other increase gradually, so that the fibres most distant from 
this surface on the one side undergo the maximum extension, and 
those on the other the maximum compression. A portion of the 
body A K B, Fig. 334, bounded before the flexure by the cross- 
sections K L and N 0, assumes, in consequence of the flexure, the 
form K L Ox JVj, by which the cross-section N becomes Ni Oi, 



§214.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 411 



Fig. 334. 



that is, it ceases to be parallel to X L and assumes a position per- 
pendicular to the neu- 
tral surface R S. The 
length K N oi the up- 
permost fibre becomes, 
in consequence, K iVi, 
and that of L the 
lowest fibre becomes 
L Oi. The increase in 
length of the former is 
therefore JS/'Wi, and the 
decrease of the latter is 
Ox, while the fibre 
E Sin the neutral sur- 
face retains its primi- 
tive length unaltered. 
The intermediate fibres, 
such SisT U,V W, etc., 
are increased or dimin- 
ished in length becom- 
ing ^ CT;, F Ifi, etc., and 
the amount U Ux, W Fi, 
etc., of the increase or 
decrease is determined 
by the proportions 
UUx_ su 

WWx_SW 
0x~ 8 0'^^' 
Let us assume the length of the fibre 

RS= KN= L0 = unity (1), 
and let us denote the extension or compression of the fibres, which 
are situated at the distance unity (1) from the neutral surface, by cr, 
then we have for a fibre, which is situated at a distance S U or 
S W = z from this surface, the extension or compression 

U UxOr W Wx = az. 
If the body is but little bent, so that the limit of elasticity is 
nowhere surpassed, we can put the strain on the different fibres 
proportional to their extensions, etc., and we can consequently as- 
sume, that these strains are proportional to their distance from the 
neutral surface, as is represented in the figure by tlie arrows. 




412 GENERAL PRINCIPLES OF MECHANICS. [§215;. 

If the cross-section of a fibre is = unity, we have in general the 
tension upon it ~ o z B; and if the cross-section of the fibre = F, 
the tensile or compressive strain is expressed by the formula 

S = ozFU= a E .Fz, 
and its moment in reference to the point 8 upon the axis is 
M=z.GzFF= a z" FE = a E . F z\ 

§ 215. Moment of Flexure. — The tensile and compressive 
strains in the cross-section Nx Ox balance the bending force P at 
the end A of the body A B. "We can therefore apply to these 
forces the well-known laws of equilibrium. If we imagine that 
there are in action at 8 two other forces + P and — P, which 
are not only equal but also parallel in direction to the given force 
P, we obtain 

1) A coiijfle (P, — P), which produces the flexure or bending 
around 8, and 

2) A simple shearing force 8 P= P, which tends to cut off the 
portion A 8 of the body in the direction '8 P or A P. The latter 
force can be decomposed into two components Pi and P^, whose 
directions lie in the plane of the Gross-g(ection A^i Ox and in the neu- 
tral axis 8 P. If a is the angle form-fed by the cross-section JVi Ox 
with the direction ^ P of the bending force, we have 

Px = P coSy a and 
P2 = Psifi. a. 

In ordinary cases in practice tiie flexure of the body and also a is 
so small, that we can put sin. a = and cos. a = 1, and consequently 
we can neglect the component Pj? which tends to tear off the por- 
tion A 8 at Nx Ox, and, on the contrary, we can put the force Pj, 
which tends to rupture by shearing the piece A 8m Nx Ox, equal 
to the bending stress P. 

If F denote the area of the cross-section Nx Oi and^2 the modu- 
lus of rupture for shearing, the shearing force is determined by 
the product F K<i. 

If we are considering a long prismatical body, P is generally 
so small a portion of F K^ that rupture by shearing can scarcely 
occur, and for this reason it will be considered in particular cases 
only. {8ee the following chapter.) 

Since one couple (P, — P) can be balanced only by another 
couple, it follows, that the tensile strains on one side form with the 
compressive strains on the other another couple {Q, — Q), and 
that the moments of the two couples must be equal. If F^, F<i, P3, 
etc., are elements or infinitely small portions of the entire surface 



tj 315.1 ELASTICITY AND STRENGTH OF FLEXURE, ETC. 413 



F of the cross-section N = N, Oi, and if the distance of these 
portions from the neutral surface or 8 be denoted by z^, z^, z^, etc., 
the strains in these elements are 

a E . F,z„o E . F^z^,o E , F-i z^, etc., 

and their moments 

o E .F, z,\ a E .F, z,\ a E . F, z,% etc. 
Since these forces form a couple [Q, — Q), their sum 

G E {F^z^ + F., z., + i^3 ^3 + • • •)? and consequently 
F^ Zi + F^ Z.2 + F^Zz + . 



Fig. 334 a. 



must be = 0. 

But this sum can 

only be — 0, when the 

]■ aint S of the axis co- 

iicides with the centre 

i gravity of the sur- 

I ice F = F^ -V F, + 

/ 3 + . . . ; consequently 

' \e neutral axis of a lent 

I )dy passes through the 

ntre of gravity S of 

/s cross-section F. The 

moment of the couple 

< ft - Q\ 

a E {F, z,' + F, zl 
+ F, zi + ...), 
should now be put 
equal to the moment 
of the couple (P, — P). 
If we denote the dis- 
tance S H of the cen- 
tre of gravity S from 
the direction A P of 
the bending force by x, 
we have the moment 
of the latter couple =-• 
P X, and therefore 
Px=: a E{F, z,^ 
■\- F,z,^ + ,, .). 
Finally, we have for the radius of curvature MR = M S of 
the neutral surface the proportion 

MR _ S U 
R S~ UU,' 




414 GENERAL PRINCIPLES OF MECHANICS. [§216. 

or, substituting MR = r, BS=1, SU=1 and U Ui = o, 

r_l 
1~ a 

Consequently r o = 1 or a = -, whence the moment of force is 

Px = — {F, %^ + F,z,' + ., .)• 
The radius of curvature at S is therefore 

The expression F^ Zx + i^2 ^2^ + . . . is dependent only upon 
the form and size of the cross-section, and can therefore be deter- 
mined by the rules of geometry. We will hereafter denote it by 
W and we will call the quantity corresponding to it the measure 
of the 7noinent of flexure, and W ^ the moment of flexure itself 
(Fr. moment de flexion ; Ger. Biegungs-moment).* 
From the above, we have for the radius of curvature 

_ WE^ 
^ ~ Px' 
and we can assert that the radius of curvature of the neutral axis 
of a deflected tody is directly proportional to the measure W of the 
moment of flexure and the modulus of elasticity E, and, on the con- 
trary, ioiversely projoortional to the moment P x of the force. 

The curvature itself, being inversely proportional to the radius 
of curvature, increases with the moment P x oi the force, and 
decreases, when the moment of flexure W E increases. 

§ 216. Elastic Curve. — If we have determined the moments 
of flexure W E for the cross-sections of the bodies, which generally 
occur in practice, we can determine by means of these values the 
curvature and from it the form of the neutral axis or of the so- 
called elastic curve. The equation 

„ ^^ WE 

P xr = W E ov r — -^ — 

P x 

indicates, that in the case a prismatical body the product of the 

radius of curvature and the moment of the stress is constant for 

all parts of the elastic curve A B, Fig. 335, and that consequently 

r becomes greater or less as the arm x of the force is diminished or 

increased, or as the distance of the point 8 considered from the 

end A of the neutral axis is less or greater. At A we have :«; = 0, 

and consequently the radius of curvature is infinitely great ; at the 

fixed point B, on the contrary, 2; is a maximum, and the radius of 

curvature is therefore a minimum ; hence the radius of curvature 



* Moment of flexure is also used for the bending moment P x. — Tr, 



216.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 415 



increases by degrees from a certain finite yalue to infinity, when 
we proceed from the fixed point B to the end A. 

If we divide a portion A 8 oi the elastic curve, the length of 
which is = s, into equal parts, and erect at the end A and at the 
points of division 8i, /S's? ^3? etc., perpendiculars to the curve, they 
will intersect each other at the centres M^, M^, M^ of the osculatory 
circles, and the portions cut of£ Mq A = Mq Si, Mi Si = M^ S<i, 

M^ Si — ifs S^, etc., are 
Fig. 335. j^^ required radii of 

curvature r^, rg, r^ of 
the elastic curve. (See 
Introduction to the 
Calculus, Art. 33.) If 
n is the number of di- 
visions of this line, we 
have the length of a di- 
s ^ 




vision 



n 



and if we 



denote the length of 
the arc (for the radius 
= 1) of the angles of 
curvature A Mo Si = 
6i% Si Ml So = d,% 
S, M, Ss = 6,% etc., by 
6 1, ^25 ^a? etc., we can 

put - = dj 7*1 = (^2 ^2 = 

^ n 

dg rs, etc., whence we 



obtain 6. = , 6n = , So 

n ri n r^ 



nr 



, etc. 



If we suppose the elastic hne to be but slightly curved, we can 
substitute for the divisions of the arc their projections upon the 
axis of abscissas A X perpendicular to the direction of the force, 
I.E. we can put A Ki = Hi Si = Ki K^ = K^ K^, etc., so that the 
arms of the force in reference to the points Si, S^, S^j etc., are 

s 



Hi Si = 



n 



s 



H,S, = Hi Si + SiL, = 2-, 

n 

Hs S3 = H^ S^ + Si L3 = 3 -, etc, 

n 



416 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 216. 



and consequently the corresponding moments of the force or the 

values for P a; are Fs2Fs3Fs . 

— , , etc. 

n n n 

Substituting successively these values for P x in the formula 
r = -p — for the radius of curvature, we obtain the following series 
of values for the radii of curvature 



Ti = n 



WE _ 



n WE 



n WE 



2 Ps'^'" 3 Ps 



, etc.; 



hence the corresponding angles which measure the curvature are 
. ^ _s_ _ Ps' X _ _i_ ^ o ^^' 

' n r,~ n' WE' ' " n r, n' WE' 

^'- nl^,-"^' n' WE' ^^^; 
Summing these angles, we obtain for the angle of curvature 

A S — (f)" of the entire arc A S = s = x 

z= (5j + ($2 + ^3 + . . . 4- (5„ 

P s^ 



n 



WE' 



or, smce we know that 1 + 2 + 3 + 

P s' Ps' 



w 



2 



n' 



WE 2 WE' 



Fig. 336. 




+ ^ = ^, we have 



for which we can write, 
according to the above 
supposition, 

P x' 
^"^ %WE' 

This arc or angle 
(since the angle be- 
tween two lines is equal 
to that between their 
normals) is equal to the 
angle S T U included 
between the tangents 
A T Qjidi S T to the 
two points A and S 
or to the angle, which 
expresses the differ^ 
ence between the in- 
clination of the curve 
to the axis in A and in 
■^0 S. If V.' pass from the 



§217.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 417 

undetermined point S to the fixed point B, we must substitute 
instead of s the entire length I of A S B, or approximately the 
projection A C of the same upon the axis of abscissas, and under 
the supposition that the curve at B is perpendicular to the direc- 
tion of the stress or parallel to the axis of abscissas, the angle (p 

becomes A I) B = fi = ^ ^^ 



2 WB' 

and, on the contrary, the angle of inclination or tangential angle 
TSII= S TX^hecomes 

_ pr _ Fs' _ p (r - s') _ p (r - x') 

a-fi 'f'-2WB 2WB~ 2WB ~ 2WB' 

If the curve at the fixed point B is not perpendicular to the 

direction of the force, but inclined at a small angle a^ to the axis, 

we will have ^ . P T -i ^ i, n 

(3 = tti -{■ , and therefore 



fl = ttj + 



2 WB 

P (V - x") 
2 WE 



§ 217. Equation of the Elastic Curve. — By the aid of the 

latter formula we can now deduce the equation of the elastic curve. 
The ordinate of the curve X S = y is composed of an infinite 
number (n) of parts, such as Ki Si, L^ iS's, X3 8z, etc., which are 
found by multiplying an element of the arc 

A Si = Si Si = Si Ssf etc. = - 

n 

by the sine of the corresponding tangential angle 

Si A Ki, Si Si Li, Sz Si X3, etc. 
Hence we have 

KS= KiSi-\- LiSi-\- LzSi-\- .,,,0Y 
g 
y = - (sin. SiA K + sin. Si Si Li + sin. SsSiLs + ,, .). 

Substitutiug the abscissa A K = x instead of the arc ^ S = s, 
and for the sines the arcs calculated from the formula 

p (r - x') 

2 WE ' 

X 2 X ^ X 
and introducing instead of x successively -, — , — , etc., we obtain 

-' n n n 

'=I-»^[''-©'*'--(Vt-'--f~)"— 



-fe')'} 



+ r- - 



Now we have f-\-V^-...-\-V = nV and 

27 



418 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 317. 



\7il \n ) \ n ) ' " \ n ) 

(see Ingenieur, page 88), whence 



2^ = 



or 



n'^ WE 

y ~ 2 WE ' 
which is the required equation of the elastic curve, when we suppose 
that the curvature is not very great. 

If we substitute in this equation x =^ 1,^q obtain instead of y 
the height of the arc or the deflection 



B C=a = 



Pf 



3 W M 

While the tangential angle a increases with the force and with 
the square of the length, the deflection increases with the force and 
with the ciibe of the length. 

The work done in bending the body is determined, since the^ 

force 

3 WEa 



P = 



r 



increases gradually with the space described and its mean value is 

WEa 



X p — 3 

2 -"- — -2 



by the expresssion 



r 



r , T, 3 WEa' ^ P' V 

L = I P a = % — ^ 



r ~ ^ WE' 

If a girder ABA, Fig. 337, whose length is I, is supported at 
both ends and acted on in the centre ^ by a force P, the ends are 

Fig. 337. 




bent exactly in the same way as in the case just treated, but in 
this case we must substitute for the force acting at ^, \ P 



218.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 419 



and for the length of the arc A B = ^ A A — ^l. Consequently 
the equation for the co-ordinates A K = x and K S — «/ is 



y = 



Px{\ V - \ X-) _ P X (3 P - 4 x') 
4 WE ~ 48 WE ' 



I 



so that for ^ = J. C = ^ the deflection is 



y^BC = a,^ 



PI' 



1 



PV 



48 WE~ ^^'3 WE' 
I.E., one sixteenth of the deflection of a girder (Fig. 333) loaded at 
one end with an equal weight. 

If in the first case the elastic curve A B, Fig. 336, is inclined at 
a small angle a^ to the axis at the fixed point B, we must add to 
the former expression for y the vertical projection of the portion x 
of the tangent, i.e., a^ x, so that we have for the ordinate 



and for the deflection 



a 



r 



{a, 



+ 



+ 



2 WE > 



X 



P? 



3 WE 



v 



^V 



(§ 218.) More G-eneral Equation of the Elastic Curve. — 

A more exact equation of the curve A 8 B, Fig. 338, formed by 
the neutral axis of a deflected beam, can be deduced in the follow- 
ing manner by the aid of the calculus. 
If we substitute in the general equa- 
tion of § 216, W E = P X 7' the value 
of radius of curvature {from Art. 33 of 
the Introduction to Calculus), 

cU' 

d x^ d {tang, a) 
and in the latter, according to Art. 32, 

ds ~ V 1 + {ta7ig. ay . d x, 
we obtain 
P xd x\\. + {tang, of] I 
a. 




WE = 



d tang. 

When the girder is but moderately deflected, the angle a formed 
by the tangent with the axis of abscissas is but small, and we can 
therefore write 

[1 + {tang, a)'] I =: 1 + j {tang. a)\ 
and consequently 



420 GENERAL PRINCIPLES OF MECHANICS. [§218. 

rrr rr _ P X [1 + ^ (tanff. af] d X 
~~ d {tang, a) ' 

or inversely 

Pxdx d tana, a ^^ „ ,, \on , ,, 

-vfT^ET = — -, , o/^ v^ = -U- - i {tang, ay] d (tang, a), 

WE 1 + 1 {tang, ay ^ ^ \ ^^ / j v i/ / 

From the latter we obtain 

/P X dx P f* 

'WW ^~ J ^ ^^^^^' ") "^ 2 y {iang. af d (tang, a), 

or, according to Art. 18 of the Introduction to the Calculus, 

P x"^ 
^ TT^ rp = — tang, a + ^ [tang, ay + Con. 

But at the vertex B the curve is parallel to the axis of abscissas 

and a = 0; substituting, therefore, the projection C A = J of the 

elastic line on the axis of abscissas, we obtain 

P¥ 
^ TTT jyy — ~ tang. + ^ (tang. 0)' + Con. — -h Con. 

Subtracting from this the former equation, we have 

2 j^j^ = tang, a- I (tang, ay, 

or inversely, for the tangential angle 8 T N = a, 

tang, a = ^ ^^ + J (tang, ay 

~ 2 WE ^ ^ 8 W' E' ' 

t^ . P(b' - x') /, P'(b' -xy\ 

I.E., 1) tang, a = ^^^ (l + g ^. ^. j. 

d 1/ 
But tang, a = -^, hence we have 

id/ Jb 

ay-(i^ PlSl_-_^] P(b'-<^')dx ^ 

y = %WE (/(^^ - ^^) ^^ + ^w-wS^^' - f ^ ^^) 

yfvdx - J?> V x" dx + ys V x' dx - J x' dx\\^ 



+ 



P' 
8 W' E' 
P 



2 WE 
+ Con, 



§318.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 421 
Since for x — 0, y — 0, we have also Con. = 0, and 

At the vertex x = b and y is the deflection C B — a, and 



therefore 

From d s = V 1 + {tang, ay . dx = [1 + h (tang, a)^] d x 'we 
obtain, by substituting tang, a = — ^^ttT'eT"^^ 

^ - e/ r + 8 • Jp"]^^ / ^ ^ 

= I dx + -^ / ( JV OJ — 2 J'^ a;^ c? ^ + 2;* i?:?;) 

_ P^ / 2^»^2;^ :?;'\ 

I.E 5 the length of the arc 

If we assume ^ = Z», we have the total length of the girder 
Inversely we have 

and therefore 

^ Pf I _ Pn' V I 3 P' I' \ 

^ 3 WE \ 15 W EV r + 35 • ^2 ^2/. or 

^- _ziL (1 _ 3 ^' n u + 3 _ziiL\ 

3 ir.£' \ 15 W EV V ^ 35 • 1^2 ^2/j 

Neglecting the members containing the higher powers of 

P 

^^, we obtain, as m the last paragraph. 



422 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 319. 



for 



tang, a = ^ ^^ and 7/ = ^ ^^ (^ - ^ x'), therefore 

FT p r 

0, tang, a = ^^f^> and {oYx = b = l,y = a = 



X = 



3WJE' 




§ 219. Flexure Produced by two Parallel Forces. — If a 

girder A Ai B, Fig. 339, 1, and II., fixed at one end, is bent by two 

forces P and P^, whose points of 
application A and A^ are at a dis- 
tance I from each other, while the 
point of appUcation Ai of the force 
Pi is at a distance A^ B =: li from 
the fixed point B, the moment of 
flexure at a point S of the portion 

^ ^1 is 

M ^ Px, 

and, on the contrary, that of a 
point Si in the portion v4i P is 

J/i = P (? + ^i) 4- Pi ^1, 

in which 2; and Xi denote the ab- 
scissas A Xand A^ K^. 

In order to obtain a clear idea 
of the manner in which these moments vary, we can lay off, as in 
II., their different values for the different points as ordinates, e.g., 

M = y = K L, M^ = y^ =z Ki Li, and join their extremities P, Pi 
etc., by a line A L H L^ G^, which will limit the values of M and 
Ml for the whole length of the beam. 

If the girder were subjected to the force P alone, the line 
bounding all the values of if or ^ = P :^ would be the straight 

line A G, the ordinate of the extremity G of which is B G = 

P . A B = P (l ~[- li). By the addition of the force Pj the por- 
tion H Goi this right line is replaced by the right line H Gi, whose 

extremities P"and Gx are determined by the co-ordinates A Ai =1 

and AiH = P I, and also A B = ? + /, and B Gi — B G + 

WGi = p{i-hi,)+ p, Zi. 

If the force P is negative, the moment M — y — Pa; of a point 

Xupon A Ai= I remains unchanged, while, on the contrary, that 
of a point Ki upon A^ B becomes M^ = y^ = P (I -^ Xi) — Pi Xi, 
and the moment of flexure at the fixed point P is = P (Z + Z,) — 



§219.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 



423 



Pi Zj, and it is positive or negative as P (Z + li) is greater or less 
than Pi li. In both cases the moment of flexure decreases grad- 
ually from Ai, remaining in the first case, Fig. 340, positive, and, 

Fig. 340. Fig. 341. 





on the contrary, in the second case. Fig. 341, becoming = for a 

P I 
point at a distance A^ = Xi= ^ ^ from A^, for greater 

values it takes the negative sign, and at the fixed point B it is 

= -[F,l,- P {I + k)l 
In the first case the right line H Gc^, Fig. 340, II., which repre- 
sents the moment of flexure at a point K^ between A and P, passes 
below the base line A B and ends at a point G^, whose ordinate is 

B Gi = P {I -{- li) — Pi li. In the second case, on the contrary, 

the right line II Gi, Fig. 341, II., rises from the point above A B, 

and the ordinates become Ki Li = y^ = — [Pi x^ — P {I + x^)'\ 

and B G, = a, = - [PJ, - P (I + l,)l 

WE 
Since the radius of curvature r = —j^ of the girder is inversely 

and consequently the curvature itself is directly proportional to the 
moment of flexure M, the graphic representations in II. of figures 
339, 340 and 341 furnish us also a representation of the variation 
of the curvature of the girder. In the case represented in Fig. 339, 
where the forces P and Pi acting upon the girder have the same 
direction, the curvature increases gradually in going from A to P, 
but if P and Pi have opposite directions, it decreases again grad- 
ually as we recede from Ai. 



424: 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 2ia 



If, as in Fig. 340, P^l^ <, P {I + k), the beam is bent in one 
direction only ; but if P] /i > P (/ 4- ?i), there is no flexure at the 
point A and also at a point 0, Fig. 343, where a point of inflection 
is formed {see Art. 14, Introduction to the Calculus), and from O 



Fig. 343. 





towards B the curvature of the girder gradually increases in the 
opposite direction. If in the second case, Fig. 342, the forces P and 
P\ are equal, for a point K^^ between A^ and B, 

M^ P[l + X,) - P x,= PI 
is constant, and the curvatures of that portion ^i P of the girder 
is the same everywhere, i.e., the elastic curve is a circle. 

The radius of curvature of the portion A A^ \^ determined in 
aU three cases by the weU-known formula 

WE 



r 



Px' 



and that of the portion A^ B^m the first case by the formula 

WE 

'^'~ P {I + X,) + P, x^ 
and, on the contrary, in the second and third cases by the formula 

WE^ 

^' ~ p (? + X,) - Pi x; 

WE 
When, in the second case, P] = P, r^ becomes = -p-y- or con- 
stant, and in the third case, where P^l^y P (I + h), for the point 

PI 
0, whose abscissa x^ — p^^^p, we have rj = qo (infinitely great). 



§220.] ELASTICITY AND STRENGTH OF FLEXURE, ETO. 425 



WE 
and, on the contrary, for the point Ai, r = -p-jy and for the point B, 

WE 

According as P Z is greater or less than Pi li — P (I + ?i) etc., 
I. E., P ^ n, in the latter case we have r ^ ri or the curvature at 
Ai greater or less than that at B. 

§ 220. The Elastic Curve for Two Forces. — The equa- 
tions of the elastic curve, formed by the axis of a girder subjected 
to the action of two forces P and Pi, can easily be deduced from 
the formulas found in paragraphs 216 and 217. 

If a denote the angle of incli- 
nation of the elastic line at A^, we 
have first for the portion of the 
curve A A^, Fig. 344, I, the arc 
measuring the inclination of the 
same at 8 

1) a = a^ -f- 



2 WE ' 

and the ordinate K 8 corresponding 

to the abscissa A K = x 

P x{f - { x") 




2 WE 



%) y — a^x + 

(compare § 217). 

By putting x = Oin (1), we deter- 
mine the angle of inclination in A 

PV 



a„ 



a. 



2 WE' 

and, on the contrary, by putting x — Hn (2), we obtain the ordi- 
nate at ^1 . ^ ^ P f 

A, O ^- a = a, I + -^-.^. 

For a point in the second portion of the girder A^ B the mo- 
ment of flexure P {I -\- x^) + P^x^ = P I + {P + P^) x^ is com- 
posed of the two parts P I and (P + Pi) x^, one of which, being 
constant, bends this portion of the beam in an arc of a circle, 

WE 
whose radius is r = -^-y- and whose angle of inclination at a point ' 



PI 



/^i situated a distance Ai 8i = Xi from A and B 8i = Ix 
B ia measured by the arc 

Zi — Xi _ P I (li — Xi) 

~~r~~ ~ WE • 



x\ from 



3, = 



426 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 320. 



The inclination at S of this portion of the girder, due to the 

flexure produced by the moment 
(P + Pi) Xi, is measured by the arc 
(P + P,) ih' - X,') 



P. 



2 WE 
§ and consequently the total inclina- 
tion at the same point is 

{P ^P,){l,'~x,^) 
"^ 2 WE 

The deflection of B S^, due to the 
curvature in a circle measured by (3^, 
is according to the well-known for- 
mula for the circle 

_ BSl _ ft ~x,y Pl ^ 
~ %r ~ 2 WE ' 




N,S, 



hence that of the entire piece B A^h 

p 7 72 

T> ri — ^ 

^^'-YWE' 

and the height of the point Si above ^i is 

K,8, = BC,- iV, S, = ^^.-^ = ^^-^ . 

According to what precedes (§ 217) the deflection Xi Si = 

^^ ^- 1^ Ij — ^ — ^ corresponds to the angle of curvature 

_ (P + Pi) ill' - Xi^) 
^' ~ 2 WE 



and the total deflection is therefore 

■ 3 ^1 ) 



A^ ;r <? - ./ - ^ ^ (^ h ^1 - ^i'') + ( P + PQ Xi ill' 

Substituting in (3) a^i = 0, we obtain the angle of inclination 3, 
which we had assumed as given, and its value is 



a, = 



2 P Z /, + (P 4- PQ li' 
2 WE 



Now if we substitute in (4) Xi = l^, we obtain by this means the 
deflection 



6 WE 
Finally, the total deflection of the whole girder is 



§ 221.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 427 



B D — a -\- ax~ a^l \ 



pv dPm + 2{p + p,)h' 



= aj -\- 



3 WU'^ 6 WB 

p ^ (2 r + 3 ^r) + 2 (P + Pi) h' 

6 W^U 

P (2 r + 3 / h' + 2 l{') + 2 Pi /i^ 



Fig. 34.5. 



6 r^ 

If the beam ^ P is not horizontal at P, but inclined at a cer- 
tain angle fi^, we must add in (3) /S^ to (3, and in (4) to y^, [3^ x^. 

If the force Pi acts in an opposite direction to P, we must sub- 
stitute in the fundamental formulas (3) and (4) P — Pi instead 
of P + Pi. 

§ 221. Girders Supported at One End. — The formulas 
of the foregoing paragraph are applicable to numerous cases in 

practice. If, for exam- 
ple, a girder A P, Fig. 
345, is at one end im- 
bedded in a wall and 
at the other merely 
supported, the question 
arises, what is the bend- 
ing force at ^, or what 
force has the support at 
A to bear, when the 
beam is loaded with a 
weight Pi, suspended at an intermediate point A^ ? 

P is here negative, /^^ = and, since A and B are at the same 
level, the sum of the deflections G Ax^=^ a and d B =z a^, is = 0, 
/ PI' 

I.E. Icii 




,)^ + iZ!iil±_y|jzZiiii! = o, 



or since Oj — 



^ 3 W E> 

WE 



WE 

I 

5 we have 



P-Pl, ^-\{P-P,)l,H + \PV + hPll,' + 1{P - P,)la' = 0. 
Prom this it follows that 

(3 ? + 2 ZQ ?i' Pi 

f + ^{VU +11') + ^i' 2' 
E.G., for I — Zi, that is, when Pj is applied in the middle of the 
girder, we have p _ ^ p 

^ - 16 ^^• 
Hence the moment of flexure at ^i is 

Pl = ^PJ, 
and, on the contrary, that at P is 



428 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 323. 



or greater than that at A^. 

Jl I = li and the points A and B are not situated upon the 
same level, if, for example, A lies a distance a^^ higher than B, wo 
must put a + ai ~ ac^. But in this case 
(3 P - PO ? 



«! = 



2 WE ' 
P l^ 



(11 P - 3 PO P 



and 



3 r^ 6 W^B 

_ [3 P + 2 (P - Pi)] r _ {5P-2P^)l\ 

^' ~ 6 WB ~ 6 Tf ^ ^ 

hence we have 

(16 P - b p,)r_ 

6WB ~^'' 
and consequently „ _ 6 W B a.2 , ^ p 

16 r 16 " 

If the moments at Ai and P should be equal and opposite, 
we must put P I = P, I - 2 P I, 

Pi 



or 



3 P = Pi, I.E. P = 



3^ 



in which case we must make 

PP 



p,r 



6 WB 18 WB' 



p^r 



If, therefore, the end of the girder lies 0,0555 -ttT^ higher than 

P I 
B, the moment of flexure in A and P is = ± — ^? or smaller than 

o 

when A and B are at the same height. 

With the aid of the values found for P we can calculate the 
radii of curvature, the tangential angles, etc., of the portions A A^ 
and ^1 P of the curve. 

§ 222. Flexure of a Girder supported at both Ends. — 

Another case, to which the formulas of the last paragraph are 
applicable, is that of a girder A P, Fig. 346, supported at both ends 

A and B and acted 
upon by a force Pi, 
whose point of ap- 
plication ^1 is at a 
distance I from one of 
the points of support 
A, and at a distance 
li from the otheTo 



Fig. 346. 




§223.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 429 

Here the moment 



P , B A = the moment Pi . B Ai, 
LE. P{1 -h ?i) - Pi k, 

and consequently the pressure on the point of support A is 

and, on the contrary, the pressure on the point of support B is 

Since A and B are situated in a horizontal plane, we have 

a + ai = 0, 
and the angle 13 is not here = 0, but is a negative quantity G B Ti 
to be determined. 

We have here 

and also 

a ~ 3 7 4. iPlh' + UP-P^)^^' 

«1 — — P fl + ypr-^ , 

and therefore their sum 

or 

6)8(2 + y Tr^=P(2Z^+6rZi + 6Z/,'' + 2 Zi^) - P^ (3 H,' + 2 Zi') 

= [2r + 6rii-{- QW +2Zi^-(3ni + 2?,^) (z + y] p, 

from which we deduce the angle of inclination at B 

^ ~ 6(1 + k) WB ~ 6(1 + l,y WE 
and that at A 

_ p,ii,{r + dii, + 2 1,') 
^~ 6 (i + uy WE 

If, for example, Pi is suspended in the middle, we have 

?i=?andP= e = y, 
and therefore 

^ " ¥w~E =" rw~E (^^^p^^^ § ^1^)- 

With the aid of the angle i3, thus determined, all the relations 



430 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 323. 



of the flexure of the girder can be determined by the formulas 
found in what precedes. 

The maximum value of the moment of flexure is for the point 
of appHcation A-^, and it is ' 

and it is a maximum for I = l^, I.E., when the weight is hung in 
the middle, its value is then 



M =: ^ll-^-Ill = 1P,1 



1-: 



223. A uniformly loaded Grirder. — If the load is uniformly- 
distributed oyer the girder A B, Fig. 347, and if the unit of length 

bears a weight = q, or the whole 
girder, whose length is I, bears the 
load Q = I q, and a portion of the 
girder A 8 = s the load q s, we must 
substitute, instead of the moments 

12 3 

- P s,- P s, -P s, etc., the moments 
n n n 

for the centres of gravity of the loads q \~),q\ — )^^\ — ) ^*^-> 




lie in the middle of 



s2s35. TXT- , s . ^ s 

-, — , , etc., and their arms are ^ -, i — , 



3s 

n 



, etc. In this way we find the angles of curvature of the ele- 



ments of the arc 



6,= 



q s' 






^_!jLlf!. A _ 1 ^' • g ^' 



'^' n' W B'^'~^'n' WE' ' ~ - ' n' WE' 



etc., 



and therefore the angle of curvature oi A 8 = si% 
1 c[8' 



</> 



^'n' WE 



(1' + 2' + 3' + . . . + n') = —^ - 



'-^ '^WE' approximatively = ^^, 



lix — I, we have the tangential angle TAG 
end ^ r. _ _J[l' _ Q I' 

^ ~ Q'WE ~ WWE' 
and therefore for a point 8^ whose abscissa \^AK=x, 

x'). 



2n' WE' 3 



UTB of the 



a 



^-'^=erirs('' 



223.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 431 



a = 



From the latter measure for the angle we find for an element 
of the ordinate x x q 

m' " 

2x \U3 X 

m )\m 



m 



6 W E^^ ~^'^' 
substituting instead of 7^ successively ( — 1^ ( 



>: ( ~ ): 



— h we 



obtain the required equation for the ordinate K 8 = y, 



X 

y = — 

_ ^ 



y^ 



6 WE 

* 6 WE 
qx 



["-imy^ 



(-•-I)- 



Q W E 
Assuming again i?; = /, we obtain the deflection 

_ ^ ^^ _ e ^^ _ 3 



ql 



QF 



3 }V E' 



6WE "" SWE SWE 

I.E., I of what it would be, if the load acted at the end of the girder. 
The ordinate of the middle of the girder is 
ql fp l'\_ 31 gP 



yi 



32/ 



12 WE V 32/ 12. 32 WE' 

hence the distance of this point below the horizontal line passing 

through B is _ __ 17 q P 

y.-a-y,- 1273217:^' 

and therefore the mechanical effect corresponding to the deflection 
a or to the sinking (3/2) of the centre of gravity of the load Q — Iq, 
when Q is gradually applied, is 

T - , n - 1 7 - y^ q^l__ _ n Q' r 

^ - 2 V 2/2 - 2 ^ « ^2 - 24 32 , WE ~ 24 . 32 . WE 
If the girder is acted upon simultaneously by a uniformly dis- 
tributed load Q and a force P at the end, we have the deflection 
Pr Qf _ IP_ Q\ r 



a 



+ 



\3 8/ W 



3 WE ' 8 WE V3 '8/ WE' 
If the girder ABA, Fig. 348, is supported at both ends and 
carries not only the weight P applied at its centre, but also the 



Kp+q) 



Fig. 348. 



i(p+Q) 




load Q = I q uniformly dis- 
tributed over its length, we 
find the deflection C B — a by 
substituting in the expression 
(P Q\ P 

for the case represented in 
Fig. 347, instead of P the 



432 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 324. 



P + 

pressure or reaction — ^-^ at the extremity A, instead of Q the 



Q 



2 



load — -—-, which is equally distributed upon one-half B A, and 
instead of I half the length of the girder B A = ^ A A 



^l 



In this manner we obtain 



a 



=f 



6 



16/ 8 WE —(-^+8 0) 



I' 



If P = 0, we have a 



QV 



48 WE 
, that is, when the entire 



Pig. 349. 



^'48 1^^' 

load is uniformly distributed upon a beam, supported at both ends, 
the deflection is but | of what it would be, if the load was sus- 
pended at the centre of the girder. 

The iceight G of the beam has exactly the same influence upon 
the deflection as a load Q, which is equally distributed, and there- 
fore enters in exactly the same manner into the calculation. 

§ 224. Reduction of the Moment of Flexure. — If we 

know the moment of flexure W^ E of a body A B C D, Fig. 349, 

in reference to an axis N^ N^ without the 
centre of gravity, we can easily find this 
moment in reference to another axis N N, 
passing through the centre of gravity and 
parallel to the first. If the distance H Hi 
— K Kx between the two axes is = d, and 
if the distances of the elements of the sur- 
faces jP„ F.i, etc., from the neutral axis 
iV iV are = 2;,, z^^ etc., we have their dis- 
tances from the axis Nx Ni, = d + Zi, d + z<i, 
etc., and the moment of flexure is 

-WxE= [F, (d + z,Y + F,(d + z,y +.,,]E 

= [Fi{d' + 2dzx + z,') + F,(d' + 2dz, + z.f) 4- ...]E 
= [d'{Fi + F,^ ...) +2d{FxZ,-h F,z, + ...) 

+ (F^Zx' + F2Z,' + ..,)] E. 
But 

Fx + F, + ... 
being the sum of all the elements is the cross-section i^of the 
entire body, and 

FiZ, -\- F,z, -^ .. . 
being the sum of the statical moments in relation to an axis pass- 
ing through the centre of gravity is = 0, and 

(F, z,' + F,z,' + ..,)E 




§225.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 433 



is the moment of flexure W B in relation to the neutral axis N N; 

consequently we have 

W,E = {W -{■ Fd') E,or 

W,= W + Fd' 
and inversely 

W=W^ - Fd\ 

Therefore, the measure W of the moment of flexure in reference to the 
neutral axis is equal to the measure Wx of the moment of flexure in 
reference to a second parallel axis minus the product of the cross- 
section F and the square (d^) of the distance between these axes. 

From this we see that, under any circumstances, the moment 
of flexure in relation to the neutral axis is always the smallest. 
The moment of flexure of many bodies in reference to some par- 
ticular axis can often be found very easily, and we can employ it 
to determine, by the aid of the formula just found, the moment in 
reference to the neutral axis. 

§ 225. Let C K =^ X and C L -— y, Fig. 350, be the coordinates 

of a point F, referred to a sys- 
tem of rectangular co-ordinates 

XX, YY, and let C M = u 
and C N = vhQ the co-ordinates 
of the same point, referred to an- 
other system of rectangular co- 
ordinates U U,V V, and, llnally 
let Ci^ = r be the distance of the 
point i^from the common origin 
C of the two systems of co-ordi- 
nates ; according to the theorem 
of Pythagoras we have 
x^ -^ y* =^ u^ + v'^ — r^ and also 
Fx" + Fy^ = Fu^ + Fv"" = F r\ 
If in this equation, instead of F, we substitute successively the 
elements i^„ i^a, F^, etc., of the entire cross-section, and in Hke 
manner, instead of x, y, u and v, the corresponding co-ordinates 
a^i, Xi, Xz, etc., ^1, y^, y^, etc., Ui, th, u^, etc., and v^, v^, Vz, etc., we obtain 
by addition the following formulas 

F, x^ + F,x,^ + ...-{- F, y^ + F^ y 
= Fx u,' -h F,u,' + ,.. + Fi Vx^ + Fi vi 
= F, r,' + F,r,' + ., ., 
and if we denote 
28 



-X- 




2 + 
3 I 



434 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 235. 



F^ X,' + F,x,^ + ... by 2 [Fx') 
F,y;' -\- F,y,' +..M^{Ff) 
F, u,' 4- F, u^ + ... by 2 i^Fie) 
F, v,' + F, v^ + ... by 2 {Fv") and 
i^in^ + i^^rs' + ...by2 (i^r^), 
fre have 

2 {Fx^) + 2 {Fy") = 2 {Fu') + 2 {Fv') =^ 2 {Fr'). 
Therefore the sum of the measures of the moment of flexure, in 
reference to the tioo axes X X and Y Y of one system of axes, is 
equal to the sum of the measures of the moments of flexure, in refer- 
ence to the tivo axes of another system of axes, and equal to the 
measure of the moment of flexure, in reference to the origin, i.e. 
equal to the sum of the products of the elements of the cross-section 
and the square of the distances from the axis C. 

If the cross-section A G Cx, Fig. 351, of a deflected body is a 
symmetrical figure, and if the axis X X at right angles to the 

plane of flexure is an 
axis of symmetry of 
the figure, there will 
be still another rela- 
tion between the mo- 
ments of flexure of the 
body. Let 8.,'K = x 
and K F^ == y be the 
co-ordi nates of an el- 
ement of the surface 
F^ in reference to the 

system of axes XX 
and Fl>ndlet^7V 
= v be the distance 
of the same element 

from the axis U Uj 

which forms an angle X 8 U — a with the first axis X X, we 
have then 

v = MF,- MN= MF,- KL 
= K F^ cos. K F^M — S K sin, K S L = y cos. a — x sin, a, 
and therefore 

v" ■= x" {sin. ay +, y {pas, of --%xy sin. a cos. a, 
Fi v" — {sin. ay F^ x^ + (cos. ay F^ y" — sin. ^aF^xy, and 
2 {Fv") = {sin. ay 2 {Fx") + {cos. ay 2 {Fy"") - sin. ^a^{Fxy). 




336.J ELASTICITY AND STRENGTH OF FLEXURE, ETC. 435 



In consequence of the symmetry of the figure, every element 
t\^ F^. . . corresponds to another opposite element F^, F^ . . ., for 
which y, and consequently the entire product, is negative ; hence 
the sum of the corresponding products for two such elements, and 

also the whole sum 

^{Fxy\^0, 
and therefore we have 

2 {Fv') ^ [sin. ay 2 {Fx") + (cos. af 2 (F y""), or 
W = (sin. ay W, + (cos, ay W„ 
in which W denotes the measure of the moment of flexure in refer- 
ence to any axis U U, Wi that in reference to the axis of symme- 
try XX and W^ that in reference to the axis Y JT at right angles 
to the axis of symmetry, provided tliat the axes U C^and Y I^as 

well as the axis of symmetry X X pass through the centre of 
gravity >S' of the figure. 

By the aid of foregoing formulas we can often find, from the 
known moments of flexure of a body in reference to a certain axis, 
its moment of flexure in reference to another axis. 

§ 226. Moment of Flexure of a Strip. — In order to find 
the moment of flexure of a known cross-section A B, Fig. 352, I, 
of a body in reference to an axis X X, let us imagine the cross- 
section divided by lines perpendicular to X X into small strips 
and every such strip as C ^ to be divided again into rectangular 
elements F^, F^, F,„ etc. If z^, z., z-^, etc. are the distances {Q F) of 
these elements from the axis XX, we have the measure of the 
moment of such a strip 

F,z,' + F,z.! + F^z^ + ... • 
= F^ z^ . z^ ^r F^ z^ , z<, ■¥ F^ z^ . z^ ^- . . : 
Now if we lay off in Fig. 352, II, A B sbt right angles to and 

equal to C A, and join B and 
(7 by a straight line, it cuts 
off from the perpendiculars to 
C A, erected at the distances 
(C F) = Z], Z.2, Z3, etc., pieces 
of the same length (F G) = 
Zi, z.2, z,i, etc., and F^ z^, F.^ z,, 
etc., can be regarded, as the 
volumes of prisms, and F^ z^.z^, 
F^ z^ . z.2, etc., as their statical 
moments with reference to the 




436 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 237. 



axis C. The prisms F^ Zi, F.^ z^, etc., however, form together a tri- 
angular prism, whose base \^ A B C, and whose height is the 
width of the strip A C.iJ) ; the sum of the above statical moments 
is therefore equal to the moment of the prism A B C vn reference 
to the axis X X. If we put the height G A ^=^ z and the width of 
the prism = h, we have the volume of such a triangular prism 

= \-bz\ 
and since the distance of the centre of gravity from (7 is | 2 (see 
§ 109), we have the statical moment of the above prisms, and con- 
sequently the measure of the moment of flexure of the strip C A 

W ^\lz^ .\z^ \lz\ 
In order to find the moment of flexure of the entire cross-sec- 
tion A D, we have only to add together the moments of flexure of 
the strips, such as C A, into which the entire surface is decomposed 

by the perpendiculars to the axis XX. 

The most simple case is that of a rectangular cross-section 

A B C D, Fig. 353. The strips into which the surface is divided 

are here all of the same size and form to- 
gether but a single strip, whose width A D 
= 5 is that of the entire rectangle.. If the 
height ^ ^ of this rectangle is = hy we 
have for the height of a strip 

z = Ih', 
consequently the measure of the moment 
of flexure of half of this surface is 

^ ^ \2/ 24 ' 
finally, the measure of the moment of the entire rectangle is 



Fig. 353. 



A 


L 










D 


^N 














N 










> 








] 


B 










C 





ir^2^^ 

^^ 24 



12 



§ 227. Moment of Flexure of a Girder, whose Form is 
that of a Parallelopipedon. — From the foregoing we see that 



the moment of flexure of a parallelopipedical girder WE 



12 



B 



increases with the width and with the cube of the height of the girder. 
Substituting this value for W E in the first formula 

PI' 



a = 



of 8 217. 



3 WE 

we obtain the deflection of a girder, whose cross-sectibn is rectangu- 
lar , and which is fixed at one end, 



§328.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 437 

Substituting it in the second formula of the same paragraph 

_ ^ Pf^ 
^ "" 48 W E' 
we have for a 'beam supported at hotli ends 

^ ~ VbWE' 
Inversely, from the deflection a we obtain in the first case the 
modulus of elasticity _^ P V 

~ ^aTTV' 

and in the second ^ P f 

E ~ 



4:ab If 

Example — 1) A wooden girder 10 feet = 120 inches long, 8 inches 
wide and 10 inches high is supported at both ends and carries a uniformly 
distributed load of Q = 10000 pounds ; how much will it be bent ? , 

The deflection is 

_ 5 ^ ^' _ 5 ^^^^^ • 1^^' _ ^^^^^ • ^^' _ 1^5000 
^ ~ ^ ITbWE ~ ^^ ' X. 10^ . B ~ '62 .8 E ~ 4 . E~' 

135 
Substituting E = 1560000, we have a = j — —~ = 0,216 inches. 

2) If a parallelopipedical cast-iron rod, supported at both ends, is 3 

inches wide and ^ an inch thick, and is deflected l of an inch by a weight 

P = 18 pounds placed upon it at its centre, the distance of the supports 

from each other being 5 feet, the modulus of elasticity is 

Pl^ 18.60^ 18.60^ 

E = -. — =-— = - — - — ^-— r-3 = :. = 73 . 216000 = 15552000 pounds. 

A: a oh? 4. i . 2 . {^y \ ^ 

§ 228. Hollow, Double-Webbed or Tubular Girders.— 

The moment of flexure of a hollow parallelopipedical girder 

A B G D, Fig. 354, is determined by subtract- 
' ing from the moment of the whole cross-sec- 

tion the moment of the hollow portion. If 
A B ^^ h and B C ^^ h are the exterior and 
A^ B^ = Ji and B^ d — li^ the interior width 
and height, we have the measures of the mo- 
ments of flexure of the surfaces^ Cand Ax CJ 

and 




12 12 ' 

and consequently by subtraction the measure oj 

the moment of flexure of the tubular girder 

„. b¥ — bx hx 
W = — -^- . 



438 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 228, 



The moment of flexure of the single-ivebled girder A B C D, 
Fig 355 ^^S' ^^^' ^^ determined in exactly the same man- 

j) Q ner. li A B — h and B rj = h are the exterior 

height and width, and if A B — A^ B^ = b^ and 
Bi C] = hx are the sum of the widths and the 
;^;_ [ — ijvj height of the two cavities, we have by subtrac- 
tion 



W 



l If - I, %,' 



The moment of flexure of the body A B D, Fig. 356, the 
cross-section of which is a cross, is found in a 
similar manner, li A B = l and B C — li are 
the height and width of the central portion, and 
AiB^— A B = Z*! is the sum of the widths, 
and ^j Ci = ^1 the height of the lateral por- 
tions, we obtain by addition the measure of the 
moment of flexure 



Fig. 
D- 



356. 
C 



D, 



N- 



^-9i 



-N 



A, 



\ 



W =-- 



12 



Fig. 357. 



In the same manner we can determine the moments of flexure 
of many bodies which occur in practice. Thus for a body AxBx CD, 
Pig. 357, with a T-shaped cross-section, whose dimensions are 

A B = C D =^1), 

A B - A,B, = A A, + B B,^l)„ 
A B ^ B C= hsLiid 
A D, = B C, = B C - C C, = hx, 
the measure of the moment of flexure in 
reference to the lower edge A^ B^ is = mo- 
ment of the rectangle A B (7i) minus moment 
of the rectangles A^ A and B^ C\, i.e., 




A, Bi 



^ ~ ^ • 12 



b, {2h,y bk'-hji,' 



12 3 

These moments are found by assuming each of these rectangles to 
be the half of rectangles twice as high ; for these the axis iVi iVj is 
the neutral axis. 

ISTow the surface Ai C^ D = F — lli — tx^hi and its statical 
moment is 



F ,ex 
consequently the lever arm is 



l-h.^--lJi,}A^=l{J)h^ -1,%,^)*, 



§229.] ELASTICITY AND STRENGTH OP FLEXURE, ETC. 439 

the product 

and tlie measure of the moment of flexure of the body in reference 
to the neutral axis N N, passing through the centre of gravity S, is 

__ 4 {h ¥ ~ I, h,') {b I t - b, h,) -S{bh' - h hj 

~ 12 {b h - b, Ih) 

__ {b ¥ - b, h.'Y -4.bhb,h, {h - hf 

~ 12 {b h - br Ih) 

It is also easy to perceive, that the high webbed and flanged 
girders have, for the same quantity of material, a greater moment 
of flexure than the wide and massive ones. Since this moment 
increases with the surface {F) and with the square (z^) of the dis- 
tance from the neutral axis, the same fibre is better able to resist 
the bending the farther it is removed from the neutral axis. If, 
for example, the height of a massive parallelopipedical girder is 
double the width ^, the measure of moment of flexure is either 

'^^ ~ 12 ^ ' 12 « ^ ' 

the flrst formula obtaining, when we place its greater dimension 
2 b vertical, and the latter, when it is placed horizontal ; in the 
first case the moment of flexure is four times as great as in the 
second. If, again, we replace the solid girder, whose cross-section 
is J ^ by a double webbed one, in which the hollow is equal to the 
massive part of the cross-section bi hi — b li, or ii b^lii — b h = b A, 
I.E., bihi = 2 b h, or bi = b V 2 and hi = h V 2, the measure of 
the moment of flexure for the latter girder is 

bi hi' -bh' __ b V^{h V~^Y -bW _ 3 

12 " 12 - T2- ^ ^ 

I.E., three times as great as for the first one. 

§ 229. Triangular Grirders. — The measure of the moment of 
flexure of a body with a triangular cross-section ABC, Fig. 358, 
can be found, in accordance with what has been stated in the last 
paragraphs, in the following manner. 

The measure of the moment of flexure for the prism with a rec- 
tangular cross-section A B C D is, when we retain the notations 

b h^ 
of the next to the last paragraph, = — --, and consequently that of 

12 



440 



GENERAL PRINCIPLES OF MECHANICS. 



[§329. 



its half with the triangular cross-section A B Cin reference to the 
central line iVi JVi is 

yyi - 2 12 - 24_^ 

But the line of gravity WJV of the 
triangle is at a distance I A B = Ih 
from the central line or line of gravity 
Ni iVj of the rectangle, and, therefore, 
according to § 224, the measure of the 
moment in reference to i\^ iV is 

"36 ~ • 




W 



Yx-Y 



72 



12 



The measure of the moment of flexure W of a girder with a 
triangular cross-section is but one-third of the measure of the mo- 
ment of flexure of a parallelopipedical one, the cross-section of 
which has the same base and altitude. But since the latter girder 
has but double the volume of the former, it follows, that for equal 
dimensions the moment of flexure of a triangular girder is but | 
that of a rectangular one. • 

For the axis Z^ Z^ passing through the base B C, the measure 
of this moment is 

and for the axis Z Z, passing through the edge A, 

111' ^ 4:bh ' _ ih 



W. 









Fig. 359. 



18 4 

These formulas do not require the cross-section to be a right- 
angled triangle. They hold good for any other triangle A B C, 
Fig. 359, whose base B C is at right angles to the bending force 

P ; for it can be de- 
composed into two 
right-angled trian- 
gles ^ Z) ^ and ^ (7 Z> 
^ whose bases B D — l^ 
and D C = hi form 
together the base B C 
— hoi the triangle 
A B G, ^o that we have for this triangle 




§230.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 441 



W = 3'ff h h' + 3^-^, 7l' = 3^ (J, + b,) ¥ = 



36* 



It is also of no importance whether the base B G lies above or 
below the axis, i.e., whether it is placed as in I or II. The mo- 
ment of flexure in both cases is 

when the modulus of elasticity for extension is the same as that for 
compression. The same formulas can also be employed, when the 
cross-section is a rhomb A B C D, Fig. 360, with the horizontal 
diagonal B D. li B D — hi^ the width and A C = h the height, 
we have for the body with this cross-section 

w-2 i/^V^iAi^iL^: 

12 \ 2 / 48 4 12' 

I.E., one quarter of the measure of the moment of a girder with a 
rectangular cross-section of the same height and width. From this 
it follows, that for a double trapezoid ABED, Fig. 361, the height 
of which is ^ C — B D — li, the exterior width A B — C D — h 
and the interior width E F — 5i, 



12 



(5 - 50 



A' _ (3 J + h,) ¥ 



48 



48 



Fig. 360. 
E A P 



Fig. 361. 




H C G 





§ 230. Polygonal G-irders. — The foregoing theory can be 
applied to a body witli a regular polygonal cross-section ACE, 

Fig. 362, whose neutral axis X Xis at the same time an axis of 
symmetry. Since such a polygon can be resolved into triangles, 
having a common vertex S, the determination of its moment 



442 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 330. 



consists essentially in the calculation of the moment of flexure of one 
of those triangles A S B. It we denote the side A B = B G= CI) 
of the polygon or the base of one of the triangles composing it by s 

and the altitude 8 K oi the 

Fig. 362a. „„_ n , x. i.x. 

same by fi, we nave the measure 

of its moment of flexure in ref- 
erence to the axis X X 

12 ~ l8"' 
on the contrary, this moment 
in reference to a second axis 

4 

quently the sum of these two 
moments is 

4 




1 

4 



Y Y 



is 



and conse- 



+ 4^ 



This sum holds good (according to § 225) for every other trian 
gle, and therefore, for a polygon of n sides, we have 



^^ ^^ nsh L^ s" \ F ( 



s h 



when its area n . -^, is denoted by F, 

If we designate the angle A 8 X hj a, the measure of the 
moment in reference to the axis A 8 L\^ 

— }\\ {sin. ay + 'W\{cos.aY', 
but the latter is also equal to the measure of the moment TFi in 

reference to K 8 D or X X, and therefore we have 
W,= W, {sin. af + W, {cos. a)\ 
or . Tfi [1 - {sin. a)^] = W, {cos. a)\ 

I.E. Wi {cos. a.y = W/{cos. ay, and consequently 

W, = W,. 
For an axis U U, forming an arbitrary angle X 8 U = (j) with 
the axis X X of symmetry, the measure of the moment is 

W = Wi sin.^ (p H- W^ cos."" (p = TVi {sin.^ (p + cos."" </>) = Wi. 
Now if we substitute in the above equation 

we obtain for any arbitrary axis of a regular polygon the measure 
of the moment of flexure 



g 231] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 443 



W 



= W.= W.=^{k^^Q, 



or, putting tlie radius of the polygon S A — S B = r and there- 



fore h' — r"^ J-? 



W 



4 



('• - i\ 



§ 231. Cylindrical or Elliptical Girders. — For the circle, 
considered as the polygon of an infinite number of infinitely small 
sides, 5 = 0, and therefore the measure of the moment of flexure 
of a cylinder is 



4 4 



0,7854 r\ 



For a hollow cylinder or tube, whose exterior radius is r^ and 
whose interior one is r^, we have by subtraction 

4 4 4 



II 



F r 

2 



[' * m 



in which F = n (r^^ — r^^) denotes the area of the ring-shaped 
cross-section, r = -^—^ — - the mean radius and b = r^ — r<i the 




thickness of the wall of the tube. The horizontal diameter divides 

the entire circle I) F, Fig. 363, into two 
semicircles ABB and A F B, and the 
measure of the moment for such a 
semicircle in reference to the diameter 

7T T* IT T* 

^^' - 4 ~ 8 • 

But the distance of the centre of 

gravity S of the semicircle from the 

4 T 
centre C of the circle is C S = :^— (see § 113), and therefore the 

measure of the moment for the parallel axis iV^ iV^is 

W= W,- F. C~W = W, - F. (pX 

\3 n/ 

while, on the contrary, for the semicircle, whose diameter is vertical, 



444 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 231. 



W 



n r 
~8~ 



= 0,3927 r\ 



In reference to an axis i\^ JV", which forms an angle N 8 X = a 
with the axis of symmetry C D, Fig. 364, the measure of the 
moment of the semicircle is 

W = - 7^ sm. a + 77 r I p: — 77—5 1 COS. a 
8 \8 9 TTV 

= (0,3927 sin.'' a + 0,1098 cos.' a) r\ 
Fig. 364. 




B X 

From the formula 




TT r 



''^=-r' 



for the measure of the moment of flexure of the full circle, that of 

an ellipse A B A B, Fig. 365, is easily deduced. In consequence 

of the relation of the ellipse to the circle given in Art. 12 of the 

Introduction to the Calculus, when A B^ A B^ represents a circle 

whose radius C A i^ equal to the major semi-axis a of the ellipse, 

and when the other semi-axis C B of the ellipse is represented by 

D E 
i, we have the ratio ^ -^ of the width D B of an element of the 

A ^1 

ellipse to that Z>i Ei of a similarly placed and equally high element 

of the circle 

_ BB _ CB^ __ ^ 

~ B.B," C B,~ a 
But since the moment of flexure of such a strip increases with the 
simple width, the moment of a strip D E oi the ellipse is to that 
of the corresponding strip of the circle as h is to a, and conse- 
quently the measure of the moment of flexure of a body with an 

elliptical cross-section is equal - times that of a body with a circu- 
lar cross-section, i.e. 



a 



na 



TTtt^l 



§232.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 445 

If this body contains also an elliptical hollow, the semi-axes of 
which are «i and hx, we have for this body 

If a body with a rectangular cross-section has an elliptical hol- 
low around its axis, or, as is represented in Fig. 

Fig. 366. 366, has an elliptical cayity on the side, we have 
the measure of its moment of flexure 
Ie jjr ^ ^*' ^ ^i' *i 






B 



D 



/' 



V 



\D 



B 



h and h denoting the length A B and the height 
A A = B B oi the rectangular cross-section 
ABBA, and, on the contrary, ai and bi the 
semi-axes C E and C F of the semi-elliptical hol- 
low Z^i^JS'. 
§ 232. The measure Wof the moment of flexure of a cylinder 
or a segment of a cylinder may be determined very simply in the 
following manner. We divide tlie quadrant A D of the segment 
of the cylinder A B N, Fig. 367, into n equal parts, pass 

through the points of division vertical 
planes, such 2^^ D E, F G, etc. and de- 
termine the moment of flexure for each 
one of the slices D E F G, which we 
consider to be right parallelopipedons. 
The sum of the moments of these 
slices gives the moment of flexure of the 
semi-cylinder A B, and by doubling 
this moment we obtain the moment of 
flexure of the entire cylinder. If r de- 
notes the radius C A = of the cir- 
cular cross-section A B N^, sl division D G of the arc = 



¥l3 




77 r 



= - — , and in consequence of the similarity of the triangles 



1 nr 

n'~2 2 n' 

D G H and C D K,^q have for the thickness K L of the slice of 

the cylinder D EF G = 2 D G L K 



KL^ G n= 



KD 



D G = 



KD 



IT r 
2~n 



n 



KD. 



CD ' ^ ^ CD 2 n % n 
Now according to the formula of § 226, the measure of the moment 
of flexure of the slice D E F G\& 

_ K~L.{%KDY _ 8^ _Tr_ -^-=:, ^ jt_ 
" 12 ~12'2w 3/1 



KD\ 



"446 GENERAL PRINCIPLES OF MECHANICS. [§232. 

If we put the variable angle A C D, which determines the dis- 
tance of the slice from the vertical diameter, :=: 0, we obtain the 
ordinate or half-height of the slice, D K — r cos. cj), and therefore 

the last measure of the moment of flexure can be put = (cos. (bV 

3 n ^ 

_nr' 3 +4 cos. 2 ( p + c os. 4 ^\4_3 -f 4 cos. 2 + cos. 4 

- 3^^ 8 ■ ' ^' ^'''° ^^ ^"8 

(see the "Ingenieur/' page 157). In order to find the measure of 

the moment of flexure for the semi-cylinder, we must substitute in 

the factor 3+4 cos. 20 + cos. 4 0, for successively the values 

7T TT 71 7T 

1 . - — , 2 . - — , 3 . - — , to n . - — , then add the results found, and 

2 n 2 n 2 n 2 n 

IT T 

finally multiply by the common factor . Now the number 3 

added n times to itself gives 3 w, the sum of the cosines from to tt 
IS = 0, since the cosines in the second quadrant - to tt are equal 

aad opposite to the cosines in the first quadrant to -, and the sum 

3 

of the cosines in the third quadrant tt to - tt cancel those in the 

3 

fourth quadrant - tt to 2 tt ; therefore the measure of the moment of 

flexure of the semi-cylinder is 

y ~ 24^ • "^ "" - ~8~' 
and that of the entire cylinder is 

4 

W='^ = 0,7854 r\ or 
4 

W = ^-? = 0,04909 d\ 
64 

d = 2 r denoting the diameter of the cylinder. 

(Remaek.)— If we employ the formulas of the Calculus, d (j> denotes an 

element of the arc 0, and the element I)G = ^r-~ = rd(p; hence the meas- 

ure of the moment of the element D E F G oi the surface is 

2d(j) .r\ 2 r^ d (p /S + 4: COS. 2 (p + COS. 4 <l)\ 



2r^ d (p /I 
(COS. <py = ^ ^- 



8 



= — - (3 + 4 COS. 2 (j) -\- COS. 4 (f) d (p = zrz {3 d (j) + 4: COS. 2 (t> d(l) -\- COS. 4 (j) d^)_ 
= ~[3d(i)i-2cos.2(p d{2 (f) + ^ cos. 4: (p dt^ (f) 1, " 



^238.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 447 

and consequently that of the portion A B E I) of the cylinder is 

W = -^U / d<t> + 2 / COS. 2 (l> d(2 (f) + ^ / COS. 4: 6 d {4: (p)\ i.e. 

TF = — (3 (^ + 3 sin. 2 6 -\- ^ sin. 4 f). (See Introdwtion to the Calculus, 

§ 26, I.). 

Substituting 9 = -^, sin. 2 <!> = sin. n = 0, and sin. 4 ^ = siw. 2 tt = 0, 

and doubling the result obtained, we have the measure of the moment of 
flexure of the entire cylinder 



Stt 



Trr' 



For the segment I> B, on the contrary, we have 

Tr = ^^g (3 + 3 sin. 2 <j> + I sin. 4 0) -j^ 

Ftt — 2 (j> (2 sin. 3 ^ + i sin. 4 ^ 



8 



-(- 



13 



)y 



= [Q (tt — 2 (p) — S sin. 2 (I) — sin. 4 (f] 



48* 



By simple subtraction we obtain, by means of the latter formula, the 
measure of the moment Wof a board I) JSJ F G of a finite thickness K L. 

(§ 233.) Beams with Curvilinear Cross-sections. — The 

measure of the moment of flexure W of bodies with regular curvi- 
li^iear cross-sectio7is is determined most surely by the aid of the 
calculus. For this purpose we decompose such a surface A N P, 
Fig. 368, by ordinates into its elements, and we determine the 

moments of such an element in reference to 
the axis of abscissas A X and also in refer- 
ence to the axis of ordinates A Y. 

If X is the abscissa A N and y the ordi- 
nate N P, we have the area of an element 
d F =i y d X 

(see Introduction to the Calculus, Art. 29) 
and therefore the measure of the moment 
of flexure in reference to the axis A X 

dW, = \f.dF=iy^dx 
(see § 226), and, on the contrary, that in reference to the axis A Y 

d Wi = x^ y d X, 

since all points of the element are at the same distance x from A Y, 

By integration we obtain for the whole surface A N P = F 




448 GENERAL PRINCIPLES OF MECHANICS. [§233. 

and 

Wi= I x^ y dx. 

If we have determined (according to § 115) the centre of gravity 
of the surface A N P and its co-ordinates A K = it and K 8 = v, 
we find the measures of the moments of flexure in reference to the 
axes passing through the centre of gravity and parallel to the co- 
ordinate axes by putting 

W, = l ff dx-v" F 

and 

W<i= I x^ y dx — w^ F, 

E.G., for a paraholic surface A N P, whose equation is ^'^ =^ p x, 

we have {according to Art. 29 of the Introduction to the Calculus) 

F = ^ X y, and (according to § 115) 

u = ^ X and v = ^ yy 
hence 

Since also from y"^ = p x, it follows, that x = -^ and d x = 

2 u dy , 
—^ — -. we have 

1 -^ 5 1 TTT 2 

and 

/' 2 7 ry' ^y'dy ^ r .;i ^y' ^ s 

J^'ydx^ J^,.^^^jjfdy^-^ = ^x^y 

o Z 2 77'2 

Finally we obtain 



and 



§ 334.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 449 




Fig. 369. Tor a symmetrical parabolic surface 

^ A D B, Fig. 369, whose cord J. ^ = 5 and 

whose altitude C D = h, we can put the 

measure of the moment in reference to the 

axis of symmetry X JT 

' ^ \2/ 20 30' 

while, on the contrary, that in reference 
to the axis Y Y Sit right angles to it re- 
mains 

'^^ ~ 175 ~ 175 ^' 

§ 234. Curvilinear Cross-sections,— If we are required to 
calculate the moment of flexure of a body, whose cross-section 
forms a compound or irregular figure, we must either divide this 
cross-section into parts, for which the measure W is already known, 
or we must decompose the same by vertical lines, calculate the 
measures of the moment of flexure of these strips (according to 
§ 226), and, finally, add these values together, in doing which we 
can employ with advantage Simj^son's or Cotes' rule. 

If, E.G., A B E C, Fig 370, is such a figure or such a portion of 

the cross-section of a body and if its mo- 
ment of flexure in reference to the axis 
A X is to be determined, we calculate first 
the measure Wi for the portion of surface 
A B G D and then the measure W^ for the 
part C ED\ subtracting the latter from 
the former, we obtain the required moment 

If the base A I) of the first part = x, 
and the altitudes of the same at equal dis- 
tances from each other are Zq, z^, z^_, Zz, ^4, we 
have the corresponding measure of the mo- 
ment, according to Simpson's rule, 

{z,' +4.Z,' + % z,' + ^z,' + z^). 



Fig. 370. 




W,= 



X 

12 



If, on the contrary, the width C D of the piece G D E to be 
subtracted be = x^ and the altitudes of the same are y^, y^, y^, y^, 
we have, according to Cotes' wile [see Introduction to the Calculus, 

1 .T, 



Art. 38), 



29 



3 * 8 



(2/0' + 3 «/,' + 3 y,» + ^3^). 



460 GENERAL PRINCIPLES OF MECHANICS. [§335. 

If ^ X does not pass through the centre of gravity S of the 
entire surface, we must reduce it by the well-known rule (§ 224) to 
the axis passing through S. In the same manner other parts of 
the cross-section, which lie below A X or alongside of A Y, may 
be treated. The centre of gravity S can be determined either 
according to § 124, or empirically by cutting a pattern of the 
section out of thin sheet iron or paper and laying it upon a sharp 
knife-edge. If we determine in this way two lines of gravity, their 
point of intersection gives the centre of gra"\dty. 

Example. — A B G E G,m Fig. 370, is a portion of the cross-section 
of an h'on rail, which can be considered as the difference of two surfaces 
A B G D and G E D. If the width of the first is f inches and that of the 
second 1 inch, and if the heights of the first are 

Zq — 2,85; z^ = 3,82; z^ = 3,74; z^ = 3,60; ands^ = 2,30, 
and those of the second 

2/o = 0,30 ; 2/i = 1,50 ; y^ = 1,80 and y, = 3,15, 
we have for the measure of the moment of flexure of the first portion 

^1 ^ 3 • 3 • 12 • f^^'^^' "'' ^'^^' "^ ^ • ^^'^^' "*" ^'^^'^ + ^ • ^''^^'^ 



1^ 

37 

37 



(33,149 + 13,167 + 4 . 40,003 + 3 . 30,571) 

236,47 = 8,7584, 



and, on the contrary, that of the second portion 

TFg = g . 1 . i . [0,20^ + 2,15^ + 3 (1,50^ + 1,80^)1 

1 37,5674 

= ^ . (0,0080 + 9,9384 + 27,6210) = — ^-r— = 1,5653, 

consequently, the required measure for the entire surface A B G E Ois 
W= W^ - W^ = 8,7584 - 1,5653 = 7,1931. 

Remark. — We can also put 
^ = I2 (t)'^^ . 0\ 2/o + 4 . r . 2/1 + 3 . 3^ 2/, + 4 . 3^2/3 + 1.4^2^4) 

= 192 (^ 2/i + 8 2/g + 36 2/3 + 16 yd, 

when 2/y, 2/1, 2/3? ^s? 1/4: denote the widths measured at the distances 
I- 2, i 2, f 2, I s, I s from A X. 

§ 235. Strength of Flexure. — If we know the moment of 
flexure of a body A K B, Fig. 371, fixed at one end B and at the 
other end A subjected to a force P, we can find the strain in every 
one of its cross-sections NO. US denotes the strains per square 
inch at a distance S N = e from ths neutral axis S, the strains at 

the distances Zi, ^^s; . • • . > are /S'l = — S, 8^ = — 8, and their mo- 

6 G 



335.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 451 



ments for the cross-sections F^, Fc^, . . ., are 

M, = F, St Zi = Fr z,^ .~M, = F, S, z, = F, z^ — , etc., 

and consequently the sum of the strains in the cross-section iV^ is 

M^- M,^- 31, + ... = (F, z,' + F,z.^ + ..,)^= -— . 

e e 

Now if X is the dis- 
tance S Hoi the cross- 
section N from the 
point of application A 
of the force P, we have 
also M = P X, and 
consequent^ 

W 8 



Fig. 371. 




1) P X = 



or 



Pxe^ W 8, 
and the strain in the 
body at the distance e 
from the neutral axis is 
Me _ Px e 



2) 8 



The latter increases 
with X, and is therefore 
a maximum for x = I, 
I.E., at the fixed point 
B, In like manner it 
increases with e, and is 
therefore a maximum 
for the point most dis- 
tant from the neutral 
axis. 

If the body is no- 
where to be stretched 
beyond the limit of elasticity, the maximum strain 8 should at 
most be equal to the modulus proof strength T, and consequently 

P 7 p 
we must put 8 = T - 



II 



r y 



or 



PI 



W T 



from which we obtain the proof strength of the girder A K B 



452 GENERAL PRINCIPLES OF MECHANICS. [§235. 

p_WT 

In like manner we have for the ultimate strength or force 

necessary to break the body at B 

_ WK 

^' - ~Te~' 

in which we must substitute for K the modulus of ultimate 

strength determined by experiment upon rupture. The funda- 

WE . « 

mental formula P x = , found m § 215, can be obtained 

r 

directly as follows. 

If we denote by g the extension N N^ produced by the strain S, 
we haye S = o B, and substituting in the proportion 

iv^iv; _ R_s 

SJV "^ ME' 



J^ J^i = a, S JV = e, E S = 1, and M E = r, the radius of curya- 

ture, we have - = - or cr = - ; hence it follows, that 
err 

r e r 

and therefore also 

WU WS 
P X — = -. 

,(i re 

If in the formulaic = \ 'uTp (§ ^^^^ ^^^ ^^® work done in 

T W 

bending the body A K B wq substitute the moment P I = 

and the modulus of proof-strength T = a B, we obtain 

T'' W^ I W L 

^ ^ e' ' WE~ - de'' 

But (according to § 206) h o^ E \& the modulus of resilience A ; 
therefore the work done in bending a body to the limit of elasticity 
is T - A ^^^ 

If l is the greatest width of the body, we can imagine the whole 
cross-section F of the body to be divided in n equally wide strips, 

whose width is -, and whose altitudes are Zi, Zc^, z^ .. ., and we can put 

F = - (zi + z^ + Zs + , , .) and 
n 



§336.] ELASTICITY A^D STRENGTH OF FLEXURE, ETC. 



453 



12 n 



and therefore also 

Wl = 






+ z, + Zs -\- ,. . / 12' 



We can make Zi = jt^i e, 2:2 = /^s e, 2^3 = jUg e, [ii, fi^, fi^ denoting 
numbers dependent upon the form of the cross-section, and there- 
fore we have 

J \ Wl ^ /fir' + fl,' + ^3' + . ♦ A Fl 

^ e^ \ Ml 4- jtfcs + /^3 + . . . / 



H 



+ fl, + H'Z + 

and consequently the mechanical effect 



12' 



But 






-1 ///!=' + M/ + l^s' + 






/^l + /^2 + ^3 + 



. . . / 12* 



is a coefficient \p, dependent upon the form 

1^1 + fh + N 

of the body alone, and F I = F is the volume of the body ; hence 
the work done L — ^q "(p A F is not dependent upon the indi- 
vidual dimensions, but only upon the form of the cross-section and 
the volume of the body, which is bent. When the bodies are of the 
same nature and of similar cross-sections, the work done is propor- 
tional to the volume of the body. 

For the work done in producing rupture we must put 

^' - ^ • 3~#' 
B denoting the modulus of fragility. 

§ 236. Fonnulas for the Strength of Bodies. — For a paral- 

lelopipedical girder A C B, Fig. 372, the length of which is I, the 

width h and the height h, we have 
e = ^ 7i, and, according to § 226, 

W = — TT ; hence — = -7^, the proof 
12 e 6 ^ 

strength of the girder is P = 



Fig. 373. 



lM!!lM!iS!J 



iiwTiirili! 



h ¥ T T 

V —— -— , and its moment i^Pl=l)¥ . -^. 

lb 

From this it follows, that the mechanical effect necessary to bend 

the girder to the limit of elasticity is 

3 e e~ 3 ■ 6 ^~T>At>hl-yAV. 



454 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 336. 



If the girder is liollow, and if its cross-section is shaped as is 

represented in Fig. 373 and Fig. 374, we have 

W IW - I, h' h If - h K' 1 

-, whence 



e 
P 



12 . 3^ 7^ 
6 hi 



6h 



T, 



h and h being the exterior and Z»i and hi the interior width and 



Fig. 373. 



D 



^ 



1^1 



Fig. 374 
D C 



Fig. 375. 



N 



n; 



— - -:N 



B 








height of the cross-section. For a body with a rhombic cross-sec- 
tion, snch as Fig. 375, we have 

W hh' d¥ 



P = 



48 . A A ~ 
hh' T 
I ' 24 



24 



and from this 
hh' T 



4 / '^6' 

I.E. I as great as for a parallelopipedical girder of the same height 
A C = h and width B D = b. For a girder, whose cross-section is 
a double trapezoid, such as is represented in Fig. 376, we have 
W _ (3 ^ + b,) ¥ _ (3^> + b;)h\ 
e ~ 48 . i /i " 24 ' 

hence the moment of the proof strength is 

(3 Z> + b,) ¥ T^ 
4 • 6' 



PI = 



h denoting the upper and bx the central width and h the height of 
the cross-section. 

For a girder with a regular 2n sided base, such as A D F, Fig. 
377, I and II, we have, if r denotes the exterior radius C A, s the 
length of the side A B, h the interior radius C L and F the entire 
area of the cross-section, 



F 



F 






^=-^ir'-hs')='j(^' + tVO = 



F{r' +2 ¥) 
12 



§336.] ELASTICITY AND STRENGTH OP FLEXURE, ETC. 



455 



If the neutral axis N 0, as in Pig. 377, I, passes through the 
middle of the opposite sides, <? = r; and if, as in Fig. 377, II, it 
passes through the opposite corners, 




Hence it follows, that in the first case 

PI — — - — T, and, on the contrary, in the second 

12 r -^ 

P^l — — ——TTT ' T, while m both cases 

VZ li 



F= lnsh = nh V r' - ¥ =^ I n s V r' - {^ sf. 

P . r 

The ratio -j^ of the proof strengths is = t« 

If the nunil:)er n of the sides of a polygon is uneven, as in Fig. 
377, III, we must substitute e = r, and therefore we must employ 
the first formula only ; provided always that the direction of the 
force coincides with that of the axis of symmetry. 

For a square cross-section we have s =z 2 h = r V2, F ^ «^ 
and the moment of the proof load 



P? = 



r 



^T=3-r=0,333r'2', 



and, on the contrary. 



pa = -^T 



r'V~2 



T = 0,471 r^ T, 



For a hexagonal cross-section we have 

2 li 3 V~~^ 

s — r = — -_=^, F — — —— s- = 2,598 8% and therefor* 

P I = -^- s" T ^ -^ r' T^ 0,541 r^ T, and 
lb lb 



Pa = i^ T 



%r^ T 



0,625 r' T. 



For a regular octagonal cross-section we have 



456 



GENERAL PEINCIPLES OF MECHANICS. 



[§ 236. 



F= 4:sh = 2 V^ ,r'= ^_— s' ; hence 
3 4/ 20 + 14 4^2 ^ ^ '^ 



and 

^^^^ 4(2|/T+1) 



^3 jr 



24/2 + 1 



7-' T = 0,691 r^ T, " 



3 4/ 17 + 12 4/^ "^ 3 1/ 2 + 4/^ 

For a massive cylinder, whose radius is r, we have 

— — — — = —7-, and therefore 
e 4r 4 

P Z = ^ r^ r = 0,785 r' T =i Fr.T, and 



3 4 r 



j\A,7rr'l= j'^ A V. 



But if the cylinder is hollow, we have, on the contrary, 

1 _|_ \2r/ Fr 

7- — ■ -^- T (compare § 231), 

Z 



4 /*! 



1 + 



%r 



ri denoting the exterior, r^ the interior and r = — - — '- the mean 

radius, F = tt {ti — r^) the annular cross-section of the cylinder 

and h ^=^ ri — r^ its width. 
Fig. 379. y^p ^ girder, whose 

cross-section is elliptical, 
as is represented in Fig. 
378, when the direction 
of the semi-axis G A — a 
is that of the force, and 
that of the semi-axis C B 
= h coincides with the 
neutral axis, we have 



^M-d- 




B 



^ 4 



'e^SLjL. 



I) 



Pl = 



-n d^ h 



T= \ FaT. 



Finally, for a parallelopipedical girder hollowed out on each 
side in the shape of a semi -ellipse, as is represented in Fig. 379, 
we have 



Pl = 



^'.hh' 



\h 



4 ^ _^l_^ rji 



IW - Z-nl^ a,' 
6A 



T 




§ 237.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 457 

and, on the contrary, if the cross-sections of the hollows are para- 
bolas, 

^ ^ - ih ~ Wh ^' 

h denoting the exterior width, h the exterior height, h^ the depth 
of the hollow and a^ the height of the same. 

§ 237. Difference in the Moduli of Proof Strength. — 

W T 
The formula P = — j- for the proof load of a girder fixed at one 

end A, Fig. 380, holds good only, when the extension a and the 

compression a^ of the body are equal 
Fig. 380. ^o each other at the limit of elas- 

ticity ; for under those circumstances 
only can the modulus of proof 
strength for extension * 

be equal to that of compression 
T, = o,E. 
For wrought iron this assumption seems to be nearly correct, and 
for wood approximately so, but these relations are entirely different 
in the case of cast iron ; the latter has not only a much greater 
modulus of ultimate strength for crushing than for tearing, but 
also the compression o^ at the limit of elasticity, which can, how- 
ever, be given only approximatively, is about twice as great as the 
extension tr, and consequently the modulus of proof strength T^ 
for compression is twice as great as the modulus of proof strength 
T for extension. 

In order to find the proof strength of cast iron or of any other 
body, for which there is a perceptible difference between o and cr, 
or between J' and Ti, we must first see which of the quotients 

T T T . 

— and — is the lesser, and substitute that instead of — in the 
e Bi e 

formula p _ ^ ^ 

e I 
The other half of the beam, corresponding to the greater ratio 

I — or — ), is of course not stretched to the limit of elasticity. In 

order to reduce this cross-section and consequently that of the 
whole body to a minimum and thus to economize as much mate- 
rial as possible, it is necessary, that both the halves of the girder 
shall be strained to the limit of elasticity. Therefore we must give 
the beam such a form and such a position that we will have 



458 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 237, 



T 




T, 




e 


— 


— 7 




or 




e 




ex 




ex 



'' T, - a^ 

I.E., that the ratio of the greatest distances e and Bx of the fibres on 
the two sides from the neutral axis shall be equal to the ratio of 
the moduli of proof strength T and T^ for compression and ex- 
tension. 



rp _ 

If, then, for cast iron we have ^ = — 

1 (7i 



2 (see § 211), we 



must so fashion the cross-section of a cast iron girder that — shall 

be as near as possible = 2. A triangular girder must be so placed, 

that the half with a triangular cross-section shall be compressed, 

and that with the trapezoidal cross-section shall be stretched. If 

we place one of the sides of the prism horizontal or at right angles 

/ e 2 . . 

to the force, we have ~ — zr, while in the opposite position, we 

B -L 

have — = -. 
e 2 

We can also give cast-iron girders, whose cross-section approach. 

the shape of a T (as is represented in Fig. 381), such dimensions 

p 
that the ratio — shall be equal to 2. 
e 



Fig. 381. 
B C 



B 



LN 



Let the entire height of the beam h^ A B 
= h, the width of the upper flange be B B = 
2 B 6^= h, the height of the hollow on the side be 

A D = Jh — III li, 
the width of the same be 

the height of the lower flange be 

H L =^ hi =^ II2 h 
and its projection on both sides be 
^ L N =^h = v,l), 
then the distance of the centre of gravity s of the whole surface 
from the lower edge H is 

_ 1 h W - l, \' + I, h^ 
M ^ - e, - ^ Z> A - ^., A, + i, h 

~ 2\1 — fiiVi + fic^ Vq I 

p 
(see § 105 and § 109). If we substitute — = 2 and e + Bi = hfWe 



n 



A H M H 



have B = i h and e, = | h, and therefore the equation of condition 



§337.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 459 



ih = 



h 
2 



H'l' ^1 



+ f^a' ^^ 



f^i 



+ ^2 '^i 



which, when transformed, becomes 

fi, V, (4 — 3 /a) — ^2 1^2 (4 — 3 /^2) = 1. 

By the aid of this formula, when three of the ratios fi^, v^, fj,^ and v^ 
of the dimensions are given, we can calculate the fourth. If we 
make fig = 0, we have the cross-section represented in Fig. 382, the 
moment of flexure of which has already been determined (§ 228), 
and for which we have fi^ i^i (4 — 3 f^i) = 1. 

Remark. — Moll and Reuleaux (see their work, "Die Festigkeit der 

Materialen," Brunswick, 1853) recommend for the determination of the 

most advantageous cross-section the use of a balance, the beam of which 

forms a table. Patterns of the cross-section, cut out of sheet-iron, are 

placed upon it in such a manner that the neutral axis, determined by the 

e G 
ratio — = — , shall lie exactlv above the centre of rotation of the beam. 

If the joattern has the most advantageous form, the beam will balance; if 
it does not, we must cause it to do so by cutting away portions from the 
side of the body, until the beam balances, when the pattern occupies the 
above position. 

Example 1. — If the cross-section of a cast-iron beam has the form of 

Fig. 381, and if the ratios of the heights are 

_ ^1 _ 7 
%- X~8'^2 
we have for the ratios of the width the condition 

77 v^ - 29 1^3 = 64. 
Kthe lower flange is omitted, then v^ = 0, and we have 

5i 64 



1-8 = 



1 

8' 



= 1, I.E. 



77 



; = 0,831, 



and the thickness of the web proper is 5 — 5^ = 0,169 i. 

V. / 39\ 

If, on the contrary, we make v^ = -^, we have 1 77 — ^1 v^ = 64, and 

1 



Fm. 382. 



consequently v^ = 0,887 and v^ = ~ 



0,887 = 




0,148. For h = S inches and h = 5|^ inches, h^ is 
= 7 inches, h^ = 1 inch, h^ = 5 inches and Sg 
= f inch ; so that the thickness of the upper and 
lower flange is 1 inch, and that of the vertical 
web but ^ inch. 

Example 2. — For a girder with a T-shaped 
cross-section, Fig. 382, we have found (§ 228) 

(h h' - &i h^y -4,ii^jih^(h- \y 



w = 



12 {b h- &i h^) 



460 GENERAL PRINCIPLES OF MECHANICS. [§338. 

in which we must put 

^^ -2hh — d^h^' 
hence, if one end is fixed and the other loaded, we have 

If we put h^ = fi^ h and 1)^ = v^i^ we obtain 

and therefore if the beam is cast-iron and we substitute /^j = |- and v^ = |-, 

K, E.G., ^ is = 10 and 5 = 8 inches, and consequently 
\=^.lQ = ^ = ^,Jl — \ = If inches, 

h^ = |- . 8 = 7 and 5 — 5^ = 1 inch, 
we have 

13 8.10 7. _520 
^^-70* 6 -1-31 ^^' 
If we substitute T^ = 18700 pounds, we have for the moment of the 
proof strength, which,, for the sake of safety, we should put = 150000 

520 
PZ = -— . 18700 = 463048 pounds. 

If this beam is 100 inches long, its safe load at the free end is 

150000 
P = ■ „^ = 1500 pounds. 

K the girder is supported at both ends and carries the load in the middle, 
we have 

P = 4 . 1500 = 6000 pounds. 

While in the first case the flange must be placed on top, in the latter it 
must be put at the bottom. 

§ 238. Difference in the Moduli of Ultimate Strength. — 

If we determine the moduli of elasticity and of proof strength 
by means of experiments on bending, making use of the formulas 

^ P It .^ Pie 
^ = ^^andl^= -^, 

the values found for E and T generally agree very well with those 
given by direct experiments on extension and compression, when the 
formulas 

PI P 

E = -r-^ and jT = ^^ are employed. 

But this relation is entirely different for the modulus of ulti- 



^238.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 461 

mate strength. Since we cannot consider the modulus of elasticity 
E to be constant beyond the limits of elasticity (for it decreases, 
when the extension or compression increases), and since the mod- 
ulus of elasticity for extension is no longer equal to that for 
compression, the strains in the superposed fibres are no longei 
proportional to their distances from the neutral axis, and conse- 
quently that axis no longer passes through the centre of gravity ; 
the values of e and e^ differ in that case essentially from what they 
are, when the limit of elasticity is not surpassed. 

If W denotes the measure of the moment of flexure for the 
stretched half of the girder and E the mean modulus of elasticity 
of the same, and if Wi denotes this measure for the compressed 
portion and E^ the mean modulus of elasticity, we have for the 
moment of the bending force, when the bending becomes excessive, 

Pl= WE + W, E, 

r ' 

and if we put, at least approximately, -^ = - and -^ = — ,^and 

Ki denoting the moduli of ultimate strength for tearing and 
crushing, the moment of the force necessary to break it is 

p I either = ElK^ll^ or = ^. ( ^^^ + Tf. ^.)^ 

E e El Bi 

If we again denote the statical moment of the cross-section of the 
stretched portion of the body in reference to the neutral axis by M 
and that of the cross-section of the compressed portion of the body 
in reference to the same axis by J/i, we have the force on one side 

= and on the other — — ^ — i, and since the two forces must 

r r 

form a couple, M E = M^ E^, This equation serves to determine 

the neutral axis by means of the distances e and e^. 

For a girder with a rectangular cross-section we have 

Jf = -^ and ifi = -^, 
and therefore 

From this we obtain 



462 GENERAL PRINCIPLES CF MECHANICS. [§338. 

Substituting this value in the equation e + e^ = h, y^o, have 

h V'E, . n VE 

e = — — = — := and e^ = 



ve + ve\ ve + ve; 

The measures of the moments of flexure are in this case 
W = *-f and Jf , = ^-f, 
and consequently we have 

3 r ^ ^ ^r\ (i/E + VE,Y ^ 

~ 3 r {V~E + V'E.y ' 
and therefore the !noment necessary to produce rupture is 

„, .^^ K.dh' E E, IW ^ 1^ 

P I either = —r^ — . -7-—= t=^- = —^ . K . 



3 Ee ( xTe^ \/E,y ^ ' ' VE + VE^ 



i¥ ^^ Ve 

or — —^ ill 



3 Ve ^ Ve, 

For E = Eiwe have, of course, 

P I = -*# JT. 

Por wood and wrought iron, E is really about — ^1, and there- 
fore we can write approximately 

in which we must substitute for K the smaller value of the modulus 
of ultimate strength. For cast iron, E, is much greater than E, 

and therefore P I approaches the value -^ K, K being the modu- 

o 

lus of rupture for extension. For wood we must substitute the 
mean value of the modulus of ultimate strength for crushing, 
K, = 480 kilograms = 6800 pounds, which value agrees very well 
with the results of the experiments of Eytelwein, Gerstner, etc. 

In like manner, for a wrought iron girder we must substitute 
instead of E the modulus of ultimate strength for crushing K^ = 
2200 kilograms = 31000 pounds. While under the same circum- 
stances Avood and wrought iron break by crushing, cast iron breaks 
by tearing. If for the latter A'' were about = K^, we would have 
to substitute for cast iron girders, in the above formulas, the 
modulus of ultimate strength of tearing, i.e., K = 1300 kilograms 



§•' 



239.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 



463 



= 18500 pounds ; but, according to the results of many experi- 
ments, we must put 

K = 3200 kilograms — 45500 pounds, 
I.E., about the mean value of the modulus of ultimate strength for 
tearing and of that for crushing. 

This great difference is caused not only by the difference of the 
moduli of elasticity J^ and Ui, but also by the granular texture of 
the cast iron, which precludes the supposition that the beam is 
composed of a bundle of rods. 

Many different circumstances influence the elasticity, the 
proof strength and the ultimate strength of a body, so that nota- 
ble differences occur in the results of experiment. 

The wood, for example, near the heart and root of the tree is 
stronger than the sap wood and that near the top, and wood will 
resist a greater force, when the latter acts parallel to the yearly 
rings than when it acts at right angles to them ; finally, the soil 
and position of the place where the wood grew, the state of 
humidity, the age, etc. influence the strength of wood. Finally, 
the deflection of a body, which has been loaded very long, is always 
a little greater than that produced, when the weight is first laid on. 



§ 239. Experiments upon Flexure and Rupture. — Experi- 
ments upon elasticity and strength were made by Eytelwein and 
Gerstner with the apparatus represented in Fig. 383. A B and 
A B are two trestles, upon which two iron bed-plates C and C are 
fastened, and D D \s> the body to be experimented upon, which is 



M 



N 



Fig. 383. 




464: GENERAL PRINCIPLES OP MECHANICS. [§239. 

placed upon them. The weight P, which is to bend the body, is 
placed on a scale board E E, which is suspended to a stirrup M N, 
whose upper end is rounded and rests lipon the centre M of the 
girder. In order to find the deflection produced by the weight, 
Eytelwein employed two horizontal strings F F m\^ G G and a 
scale M H, placed upon the middle of the girder. Gerstner, on 
the contrary, employed a long sensitive one-armed lever, which 
rested upon the beam near its fulcrum and whose end indicated on 
a vertical scale the deflection of M in 15 times its real size. 
Lagerhjelm employed a pointer, which was moved by means of a 
string passing over a pulley, and which showed the deflection of 
the beam magnified upon a graduated circular dial. Others, as, 
E.G., Morin, made use of a cathometer to determine the deflection. 
The object observed was a point fastened in the centre of the girder. 
In the English experiments a long wedge was used to measure this 
deflection ; it was inserted between the centre of the beam and a 
fixed support. In order that the accuracy of the measurement 
may not be affected by the yielding of the supports of the girder, 
it should rest during the experiments either upon stone founda- 
tions (Morin), or a long ruler should be placed a certain distance 
above the girder and fastened at its ends to the ends of the latter, 
but in such a manner that it cannot bend with the beam, and in 
each experiment the distance between the lower edge of the ruler and 
the centre of the deflected girder should be measured (Fairbairn). 

The manner in which Stephenson, etc., determined the deflec- 
tion and strength of tubular sheet iron girders, is shown with the 
principal details in Fig. 384. The tube A B is 75 feet long (the 
front portion being omitted in the figure), is supjDorted at both 
ends, as, e.g., in C, upon blocks of wood and its centre rests upon 
a beam D D, which is carried by two screw-jacks. An iron arm, 
the end F of which only can be seen in the figure, passes through 
the middle of the tubular girder near the bottom, and from this 
two stirrups G, G hang, to which the scale-board H Hio receive 
the weight P is suspended. Before the experiment and during 
the laying on of the weights, the entire load was supported by the 
beam D D ; when the screw-jacks were lowered D D sank and 
placed itself upon the supports E, E, while the centre of the tube 
A F, loaded with P, remained free and could assume a deflec- 
tion corresponding to the force P. This deflection was measured 
by means of a wedge. 

In order to avoid the use of very large weights in experiment- 



§239.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 465 

» 
ing upon large girders, they are generally made to act upon the 
latter by means of a lever with unequal arms. With the same 
object in view, Hodgkinson caused the force of the lever to be 



Fig. 384, 




/ ' 



applied not to the centre of a girder supported at both ends, but to 
one end of a girder, which was supported in the middle and the 
other end of which was fastened by a bolt to the foundation. 

The results of experiments, made under very different circum- 
stances and with very different kinds of materials, particularly of 
wood and iron, have shown the theory laid down in the foregoing 
pages to be correct in all important particulars. In regard to the 
rupture of parallel opipedical girders ifc was proved, that those of 
wood and wrought iron, under the same circumstances, gave way 
by crushing, and that in the case of cast iron the rupture began 
either by the exterior fibres being torn apart or by a wedge break- 
ing out at the most compressed part (in the middle). 

We can satisfy ourselves of the truth of the hypothesis, made 
in § 214, in regard to the behaviour of the fibres of a body, sub- 
jected to flexure, by making saw cuts upon the compressed side of 



^66 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 240. 



parallelopipedical wooden rods and then filling them up with pieces 
of wood, by drawing a series of lines upon the side of a beam at 
right angles to its longitudinal axis, and finally by fastening two 
thin rods to the beam, one along the extended and the other along 
the compressed side. 

§ 240. Moduli of Proof and Ultimate Strength.— In the 

following table the moduli of elasticity, of proof strength and of 
ultimate strength or of rupture, as determined by experiments 
upon bending and breaking are given. The first differ but little 
from those determined by the experiments on extension and com- 
pression ; but, for the reasons given above (§ 238), this is not 
true of the modulus of ultimate strength. The upper of the 

two quantities in a parenthesis -j y gives the value in English meas- 
ures (pounds per square inch) and the lower one the same in 
French measures (kilograms per square centimeter). 

TABLE 

OF THE MODULI OF ELASTICITY, OF PROOF STRENGTH AND OF 
ULTIMATE STRENGTH OR OF RUPTURE OF DIFFERENT BODIES 
IN RELATION TO BENDING AND BREAKING. 



Names of the Bodies. 


Modulus of Elasticity 
E. 


Modulus of 

Proof 
Strength T. 


Modulus of Rup- 
ture or of Ultimate 
Strength K (JC^). 


Wood of deciduous Trees 
Wood of evergreen Trees 

Cast Iron 

Wrought Iron . . ^. . 
Limestone and Sandstone 
Clayslate 


j 1280000 
( 90000 

j 2130000 
( 150000 

j 17000000 
1 1200000 

' 28400000 
2000000 


3100 

220 

4300 
300 

10670 

17000 
1200 


9240] 

12800 ) 
900) 

45500 1 
3200) 

32700 ) 1 
2300) 

j 5000 ) 
I 350) 



In order to determine from the value in the foregoing table the 
load, which a girder can carry securely, we must introduce a factor 



§240.] ELASTICITY AND STRENGTH OF FLEXURE, ETC 467 

of safety and substitute in the formulas for the proof strength 
already found for wood 

either instead of T, ^ Tor instead of -ffi.jo -^j 
for cast iron 

either instead of T, ^ T or instead of K, \ K, 
and for wrought iron 

either instead oi T,\ T or instead of K, \ K, 
Consequently we can hereafter put for wood 
T —^Z kilograms = 1000 pounds, 
for cast iron 

r = 510 kilograms = 7000 pounds 
and for wrought iron 

T = 660 kilograms — 9000 pounds. 
We cannot employ these values in calculating the dimensions 
of shafts and other parts of machines ; for, on account of their 
constant motion and of the wearing away of the parts, a greater 
factor of safety must be introduced, which requires us to assume a 
smaller value for T. 

If we substitute these values in the formulas 

T T T 

for parallelopipedical and for cylindrical girders, we obtain the fol- 
lowing ^r«c^tca/ /ormz^/o^s ; 
For wood 

Pl = lQ'7I)h'' = 785 r' = 98 d' inch-pounds. 
For cast iron 

P I = 1167 ih' = 5500 r' = 687 d' inch-pounds. 
And for wrought iron the greatest value 

P I = 1500 bh' = 7070 r' = 884 d' inch-pounds. 
If with Morin, and in accordance with the practice in England, 
we put for cast iron 

IT IC 

instead of T, — to — = 750 kilograms, 

and for wrought iron 

instead of j; — — 600 kilograms, 



we obtain for cast iron 

P I =. 1778 h ¥ = 8376 r' = 1047 d' inch-pounds, 
and for wrought iron the smaller value 

P I = 1422 bh' = 6700 r' = 838 d' inch-pounds. 
If the load Q is not applied at the end of the beam, but is 



468 GENERAL PRINCIPLES OP MECHANICS. [§240. 

equally distributed over the same, the arm of the load is no longer 

I, but --, and consequently, the moment being but half as great, we 
must put Ql W T ^^ ^ WT 

If the girder is supported at loth ends (Fig. 337) and the load 

P acts midway between the two points of support, whose distance 

p 
from each other is = I, the force at each end is = — , its arm is = 

Z 

- and its moment 

PI WT ^ ^^ ^ WT 

--r — and P / = 4 . 

4 e e 

Therefore, under the same circumstances, the girder bears twice 
as great a load in the second and four times as great a one in the 
third as in the first case. 

If, finally, a girder uniformly loaded along its whole length is 
supported at both ends, it is in the first place bent upwards by a 

force ^, whose arm is -, and in the second place downwards by a 

force ~r, whose point of application is the centre of gravity of one 

of the halves of the load, whose lever arm is therefore - and whose 

4 

moment is ^. Consequently the moment with which one end of 

8 

the girder is bent upwards is 

Ql _ Ql _ Ql^ 

W T 
hence we have Q I = ^ . The proof load of the girder is in 

e 

this case 8 times as great as in the first one. 

For a parallelopipedical girder we have in the first case 

T . 
P I — ]) li" ^-^ m the second 



T 

Ql — 2 h li- -— , in the third 

D 

T 

P I = 4:b h'' — and in the fourth 

D 

Ql = 8bJi'^, 
& denoting the width and h the height of the rectangular cross-section. 



§ 241.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 



469 



Example — 1) What load can a girder of fir carry at its middle, when 
its width is 5 = 7 and its height h = Q inches, and when the point of ap- 
plication of the load is 10 feet distant from the supports ? Here we have 
^l = 10 . 12 = 120 inches, and therefore, according to the above formula, 

P Z = 4 . 167 5 /i' = 4 . 167 . 7 . 81, 
and the required working load is 

4676 . 81 ^ ^g^^^ ^ ^^ ^ ^^^g pounds. 



P = 



240 



2) A cylindrical stick of wood, firmly imbedded at one end in masonry, 
is required to bear a weight Q = 10000, uniformly distributed over its 
whole length 1 = 5 feet; what should be its diameter ? We have here 

Ql = 2 — ^- 
and consequently by inversion 



V- 



10000 



1570 



1570 



= 2 . 785 



— = '7*382 = 7,26 inches. 



N^ 



and the required diameter is = 2 r = 14,52 inches. 

241. Relative Deflection. — The bending of the moving 
parts of machines, such as the shafts, axles, etc., has often a very 

bad effect upon their 
Fig. 885. working, either by giv- 

ing rise to vibrations 
and concussions, or by 
preventing the different 
parts of the machine 
from engaging perfect- 
ly. We are therefore 
in certain cases re- 
quired to determine the 
cross-sections of these 
parts of machines, not 
with reference to the 
modulus of proof 
strength, but to the 
deflection, by assum- 
ing the deflection to 
be a very small definite 
portion of the entire 
length of the body or 
part of the machine. 
We have already found (§ 217) the deflection for a prismatic 
body A S B, Fig. 385, fixed at one end B and loaded at the other 
A, to be 




470 GENERAL PRINCIPLES OF MECHANICS. [§241. 

p r 

and we can put its ratio to the length A B, which is given 

~ I ~ 3 WE' 
whence, by inversion, 

PI' = 36 WE. 

Hence we have for ^ parallelopipedical girder 

PI -ze^^E=^^^-, 

and for a cylindrical one 

Pf = 3e^E= jTTdr'E. 
4 4 

Generally a relative deflection 6 = j = ^i^ is admissible, and 
we can put 

If we substitute for wood the modulus of elasticity E = 1600000, 

we obtain 

Pr = 800 i¥ = 7540 r\ 

For cast iron we have E = 15000000 pounds, and therefore 
P r = 7500 b ¥ = 70700 r\ 
and for wrought iron E — 22000000 pounds and 
PV =. 11000 !)¥ = 103700 r\ 
On the contrary, when the deflection reaches the limit of elas- 
ticity, we have (§ 235) 

2) P I = or P I' = — , 

^ e e 

and, therefore, equating the two values of P V, we obtain 

W Tl 

i^-i- = 36 WE, 

e 

and consequently the ratio of the length I of the beam to the maxi- 
mum distance e, when both the deflection and strain reach at the 
same time their limit values 6 and T, is 

I 36 E 36 



e T a' 

hence for parallelopipedical bodies 






§341.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 471 

and for cylindrical ones 

I se I e 

- = — ^'"^ -J = i -} 
r (7 a o 

a denoting the extension or compression at the limit of elasticity 

corresponding to the strain T, 

I SO 
If - < — , we obtain from the first formula the greater value 
e o 

I SO 
for P I and if, on the contrary, - > — , the second formula gives 

the greater value for the moment of the force. Therefore for a 
given moment of force (P I) the greater dimensions for the cross- 
section are given in the first case, where the length of the body is 

J e, by the formula 

W T 
e 

(O fj\ 
) e, by the formula 

3 6* W E = Pl\ 

7 O /3 -I 

If we substitute in the ratio - = — for the limit, Q = 



e a ' 500' 

we have for all materials - = -^rpr^ — = , and, therefore, for 

e 500 o a 

wood, for which a = -— , - = 0,006 . 600 = 3,6, and more par- 
ticularly for a prismatical beam of this material 

^ "^^ ^ == 10 = ''^- 

If we assume for cast and wrought iron a = t^t^tt, we obtain for 

1500 

these substances 

- = — '—-^ — = 9 and therefore 
e 500 



The formula 






2000 2000 

is of course applicable to the normal case above, I.E., when the body 
is loaded at one end and fixed at the other. For a load equally 
distributed we must substitute (according to § 223), instead of 
P, I Q. If the body is supported at both ends and the load is sus* 



472 GENERAL PRINCIPLES OF MECHANICS. [§242. 

P I 

pended in the middle, we have, instead of P, — and, instead of I, -, 

and therefore 

•^ ^ - ^ • 2000 ^ - ^ • ~^000~- 

If the girder is supported in the same manner and the load 

5 Q 
uniformly distributed, we must substitute for P, -~, 

o 

Example — 1) What load placed upon the centre of a wooden beam, 
supported at both ends, will produce a relative deflection Q = -5^, if its 
width is & = 7, its height A = 9 inches and the distance between the sup- 
ports is Z = 20 feet ? Here we have 

800 & h' 6400 .7.9^ ^ ^, 
P = 8. —y- = -^2o7i3y.- ::= 7 . 9^ = 567 pounds, 

while in the foregoing paragraph, under the assumption that the beam 
should be bent to the limit of elasticity, we found P = 1578 pounds. 

2) How high and wide must we make a cast iron girder (the ratio of 

its dimensions being ^^ = 4), which, when supported at both ends, will 

sustain a load Q = 4000 pounds, uniformly distributed over its length, 
which is 8 feet ? Under the latter supposition, we have 

I Q Z^ =--. 8 . 7500 & /i^ 

I.E., 1 . 4000 . 8' . 12^ = 8 . 7500 -^ or 7i^ = 4* . 6, 

consequently 

A = 4 V 6 = 1,5G5 . 4 = 6,26 inches and 

h 
I = -7 = 1,565 inches. 
4 

According to the formulas of the foregoing paragraph, we would have 

qi= 'd . 1167 l h\ or 4000 . 8 . 12 = 8 . 1167 . -j, 
whence the required height is 

/i = 4 y^^ = 4 . 1,37 = 5,48 mches, 
and the required width 

h = -r = l,d7 inches. 
4 

§ 242. Moments of Proof Load. — From the formula 

T 

Pl^hh' — 
o 

for the moment of the proof load of a parallelopipedical girder we 

perceive that this moment increases with the width h and with the 

square of the height h, that the iwoof load itself 



§242.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 473 

Z 6 
is inversely proporticmed to the hngtli (!) and that the height has a 
much greater influence than the width upon the solidity of such a 
girder. A girder, whose width is double that of another, will bear 
but twice as great a load as the latter, or as much as two such 
girders placed side by side. A girder, whose height is double that 
of another, bears, on the contrary, (2') = 4 times as much as the 
latter, when their widths are the same. For this reason we make 
the height of parallelopipedical girders greater than their width, 
or we place them on edge, or in such a position, that the smaller 
dimension shall be perpendicular to the direction of force P and 
that the greater dimension shall be parallel to it. 

Since b h expresses the cross-section F of the beam, we have also 

hence the moments of the proof load of bodies of equal cross-section, 
mass or weight are proportional to their heigJit. If, for example, 

h and h are the width and height of one body and - and 3 h those 

of another body or F = -3 h — b h tlie area of both their cross- 
sections, the bodies have the same weight, when the other circum- 
stances are the same, but the latter bears three times as great a 
load as the former. 

Ifb = h, the cross-section of the beam is a square, and we can 
diminish the moment of proof load by placing the diagonal in a 
vertical position. In this case, W, as we know from § 230, remains 

h If y 

unchanged and is = — ^ = — , while e becomes equal to the semi- 
diagonal -h b V 2 = b V h Therefore we have 

T 

while, if it were laid on one of its sides, we would have P l — h^ —-, 

o 

See § 236. 

The equations for parallelopipedical girders are analogous to 
those for girders with an elliptical cross-section. We have in the 

latter case (according to § 231) W = — ~— and e = a, the semi- 
axis a being supposed parallel and the semi-axis b perpendicular to 



474 GENERAL PRINCIPLES OF MECHANICS. [§ 343. 

the direction of the force or, as is generally the case, horizontal. 
Here we have for such a girder 

4 4 

the area of the elliptical cross-section being F — tt ah. The mo- 
ment of the proof load of this beam increases, therefore, with the 
area and with the height a of the cross-section. 

lih = a = Ti^fQ have a cylindrical girder , whose radius is r, 
and the equation becomes 

TT r^ T 

4 4 

The moment of proof load of this body increases, therefore, with 
the product of the area of the cross-section and its radius. 

If the cross-sections or weights are equal, the ratio of the mo- 
ment of proof load of a body with an elliptical cross-section to that 

a 
of one with a circular cross-section is -. Therefore, we should 

r 

always prefer the elliptical to the cylindrical girder. 

This holds good for all other forms of cross-section ; the regu- 
lar form (the square, the regular hexagon, the circle, etc.) has 
always, for the same area, a smaller moment of proof load than a 
form of greater height and less width. 

Regular forms of cross-section should, therefore, be employed 
only for shafts and other bodies, revolving about their longitudinal 
axis, in which case during the rotation a continual change in the 
position of the dimension of the cross-section takes place, i.e., after 
one-quarter of a rotation the height becomes the width and the 
width the height. 

§243. Cross-section of "Wooden Grirders. — If a cylindri- 
cal girder has the same cross-section i^ = tt r'^ = J^ as a parallelo- 
pipedical beam, whose height and width is = h, we have the ratio 

i_^_r. = 1,77345. 
r 

and, on the contrary, the ratio between the moments of proof load 

Jf and M^ (M^) is in the first place, when the latter body is laid 

upon one of its sides, 

M r h ^ r ^ 

^, = 4 = 6 = 2 J- = STV = I'S • 0'56^^ = 0'8^«^' ._ 
and in the second place, when its diagonal is placed in a vertical 
position, 



§243.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 475 






r 

4 



hV^ 



= 3 . 0,3989 = 1,1967. 



Fig. 386. 




12 V2 n 

The moment of proof load of the cylinder (with circular base) 
is in the first place smaller and in the second place greater than 
that of a parallelopipedon with a square base. 

Since wooden parallelopipedical girders are cut or sawed from 
the round trunks of trees, the question arises, what must be the 
ratio of the dimensions of the cross-section of such a beam, in order 
that it shall have the greatest moment of working load ? 

Let A B D E, Fig. 386, be the cross-section of the trunk of the 
tree, A D = diis, diameter and 

AB = D E= h 
the breadth and 

A E = B D = h 
the height of the beam ; then we have 
h' + Ji' = d\ or 
h' = d'' - h\ 
and the moment of proof load is 

D b ' 

The problem now is to make 

h (d' - b') 
as great as possible. If we put, instead of h,h ± x, x being very 
small, we obtain for the last expression 

(b ±x)d' - {b± xy ^hd' -F ±{d' -dh')x -^h x\ 
when x^ is neglected. Now the difference of the two expressions is 

y= ^ {d'' -?>W)x -^ ?>l x\ 

In order that the first value shall always be greater than the 
second, the difference 

y = z^ {d' ■- ^¥)x + Zhx' 
must be positive, whether we increase or diminish h by x. But 
this is only possible when tZ' — 3 Z*' = ; for this difference is then 
= ^b x" or positive, while, on the contrary, when ^' — 3 J' has a 
real positive or negative value, ^h x" can be neglected, and the sign 
of the difference :^ (d' — d ¥) x varies with that of x. Therefore, 
putting J' - 3 J' = 0, we obtain the required width 

h = d Vl, and the corresponding height 
h=VlF^:^' = dVl', 
the ratio of the height to the width is 



476 GENERAL PRINCIPLES OF MECHANICS. [§ 243i 

- = — = = 1,414 or about I, 
I \/\ ' ' 

We should, therefore, cut the trunk of the tree in such a man- 
ner as to produce a beam, whose height is to its 
^ Fig. 387. width as 7 is to 5. In order to find the cross- 

^ V section corresponding to the greatest strength, 
^, ^.■''' \ we divide the diameter A D, Fig. 387, into three 
I - -^ j equal parts, erect in the points of division M and 
' ^ / N the perpendiculars M B and N E and join 
the points B and E, where they cut the circum- 
"^ ference, with the extremities A and D by straight 
lines. A B D E \^ the cross-section of greatest resistance ; for we 

have 

AM\AB = AB\AD2.tAAN\AE = AE\AD, 

and consequently 

AB = 1) = VA M .AD = VJdTrd = ^ V| and 

AE=h= VA N .A D ^ V'i d.d = d V^, or 

- = , which is what was required. 

1 

Remake 1. The moment of proof load of the trunk of the tree is 

and that of the beam of greatest resistance, cut from the same, is 
T - _2l 8 T 

Pl^ — d^\.^d^= VM3 ^' "" VT43 ^'' 

and consequently the beam loses by being cut 
8 4 
1 _ -==; . _ ^ 1 _ 0,65 = 0,35. 

I.E. 35 per cent, of its proof strength. In order to reduce this loss, the 
beam is often made imperfectly four-sided, i.e. with the comers wanting. 
The moment of the proof load of a beam with a square cross-section, hewed 
from a tree of the same size is 

6 ^^ 2 ' 
since the width is = height = ^ VJ = 0,707 d ; the loss is 

8 4 8 

1 ~. - = 1 =: = 1 - 0,60 = 0, 40, 

6 . 2 V2 '^ 3 TT V2 

I.E. 40 per cent. 

(Remauk 2.) In order to cut from a trunk of a tree a parallelopipedical 

beam, whose moment of flexure is a minimum, or for which ^ = ^ is as 
small as possible (compare § 241), we must have 



§ 244.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 477 

= d^ h^ — 7i^ 
as great as possible. The first dilierential coefiicient of the latter expres- 
sion in reference to 5 is 6 d^ h^ — Sh'', whicb is equal to zero for h^ = f d^, 
I.E. for 



,- d ^/S , 
h = d^^ = — ^ and 



d 



For these values the moment of flexure of the beam is a minimum (see 

Introduction to the Calculus, Art. 13). 

h Vs 
Here we have ~ = ~ = 1,7321, or about f , while above we found for 

h 
the maximum of the moment of proof load , = ^. 

This condition corresponds to the construction in Fig. 387, when we 
mal^e A 31 = I) N = ^ A D. 

§ 244. Hollow and Webbed Girders. — We have, accord- 
ing to § 228, for a holloiv parallelopipedical beam 

PF = ^^ , 

and therefore the moment of proof load is 

W T WT (b h' - h, liA T 



PI 



\li \ h 



} 6- 



If we put ~ = a and — ^ = v, we obtain 
h 

= b h (1 — w' V), 

and, since the cross-section of the body is 

F = bli — bih = b li [1 — \iv), 

\1 — l-t V / 6 

Since ^ ~ ^' "" = l-f^v-hf^y- f^'y = 1 + (^ - ^') ^ ^ 

1 — fJb V 1 — /Li V 1 — fbi V 

increases with v, we obtain the maximum value of P Z for v = i, 
and it is 

1) Pl = [l + (\^) l^]^h^ = (1 + p + p") Fhj. 
If, on the contrary, we put ju = v, we obtain 

2) Pl= (1 + fj,^) Fh^. 



478 GENERAL PRINCIPLES OF MECHANICS. [§ 244, 

In both cases we must make jw as great as possible, and there- 
fore nearly — 1. If we wish the proof strength of the girder to be 
a maximum, we must make the webs as thin as possible. Hence 
we have for /x =: 1 in the first case 

T T . 

P I :=i d F h —r = Fh 77-, and in the second case 
o Z 

T T 

Plz=i%Fh— =^Fh-^, and, on the contrary, for ft = 0, 

T 

Pl = Fh^ 
o 

In all three cases the proof load of the girder, when the cross- 
section (i^) or the weight is the same, increases with the height 
(A) ; but in the first case, where the girder consists of two flanges, 
it is a maximum ; and in the second case, where it forms a paral- 
lelopipedical tube, it has a mean value ; and in the third case, 
where it is composed of one or two webs, a minimum one. 

If, for example, a massive girder, whose dimensions are h^ and 
hi, has the same cross-section or weight as the supposed tubular 
girder, we have 

F = ii hi = h h — hi hi, i.e. 2 hi hi = b h or '^^-^ = fj, v = 



hi hi . 



If we assume -^ = V? we have u = v = V^, and therefore the 
oh 

ratio of the proof loads of the two beams is 

P ^ (1 - ^' V ) h ^ /l^i\^3^^^3^T^ , 

Pi 1 — fl V hi \1 — }2/ 

the tubular girder is therefore capable of carrying more than double 
the load that an equally heavy massive girder can, whose form is 
that of the hollow of the first girder. 

The same relations also obtain for I-shaped girders, since (see 
§ 228) the measure of the moment of flexure W is the same for 
both. These formulas can also be employed for bodies with more 
than two wehs, as, e.g., bodies with the cross-section represented in 
Fig. 388, in which case h denotes the width of the 
upper and lower rib, h the entire height A D = B C, 
hi the sum jf the widths and hi the height of the 
hollow spaces M, N, 0, P. 

The formulas for a pipe or hollow cylinder are 
analogous to those for a parallelopipedical beam. If r 
is the exterior and ri = fi r the interior radius, the 
moment of proof load of this body is 



Fig. 388. 






^244.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 479 

T 

4' 

This value increases as jtfc = — approaches unity, and therefore 

as the wall of the pipe becomes thinner. If we put jit = 1, we ob- 
tain the corresponding maximum moment of proof load 

T T 

PI = 2Fr^ = ^r-^, 

4 2 

If we compare the proof load of this tube with that of a massive 

iron cylinder, whose radius r^ = fi r = r V^, we have then for the 

latter ^ ^ ^ T ^ T . 

P^l = F r^-r — a F r -r and 

4^4 

exactly what we found under the same suppositions for parallelo- 
pipedical girders. 

We can see from the general equation 
Pl^WT^ (/'...' + i^. ./ + ■■) y^ ^^^^_, ^ ^^^, + ..)eT, 

that the moment of proof load of a body increases as the distances 
z^ ~ 1^1 e, 2^2 = |f^2 &i etc., of the portions F^, F^, etc., of the cross-sec- 
tion from the neutral axis become greater. But since this distance 
can at most be = e, those girders will have the greatest moment 
of proof load, the different portions of whose cross-section are at 
one and the same distance (the maximum one) from the neutral 
axis. Such a beam consists of tiDO flanges. Since the webs which 
unite the two flanges cannot satisfy the conditions of maximum 
moment of proof load, it is impossible to attain this maximum, and 
we must therefore content ourselves with increasing the proof 
strength of the girder by hollowing it out, by thinning it in the 
neighborhood of the neutral axis, or by adding flanges at the 
greatest possible distance from the same axis. 

The thickness, which the web must possess in order to resist the 
shearing strain, will be determined in the following chapter. 

Remark. — Under the supposition that the proof strength increases and 
decreases with the ultimate strength, the English engineers increase the 
size of that portion of cast-iron girders, which is subject to compression ; 
for that material resists compression best. On the contrary, they increase 
the dimensions of the compressed side of girders of wrought iron, as the 



480 



GENERAL PRINCIPLES OF MECHANICS. 



[§245. 



latter resists extension best. If the girders are to be supported at both 
ends, their form must depend upon the substance of which they are made. 
If the beam is of cast iron, we make the bottom flange larger than the 
other; if of wrought iron, the upper flange, or the upper part of the 
girder is constructed of two flanges, united by vertical webs, as is repre- 
sented in Fig. 388. The forms T and T, discussed in a previous paragraph 
(§ 337), are employed for cast iron. 

Example. — An oak girder 9 inches wide and 11 inches high, which has 

up to the present time shown sufficient strength, is to be replaced by a 

cast-iron girder, whose exterior width is 5 inches and whose height is 10 

inches ; how thick should it be made ? If we put the double thickness of 

the metal = x, the width of the hollow is = 5 — a;, and its height is 

= 10 — a?, and consequently we have for the hollow girder 

^1 \^ - 5g Ag' = 5 . 10^ — (5 -x) (10 — xy = 3500 x — 4.50x^ + 35 a?^ _ x\ 

7000 
hence the moment of proof load is P Z = fi^Tft (^^^^ a;— 450 x^ + 35 x^ — «*). 

K the moment of proof load of the massive wooden beam is P Z = — ^— . 

9 . 11^ = 1 . 1089000, we must put 

700 . (2500 X — 450 «^ + 35 «^ — x') = 1089000, or 
3500 X — 450 x^ + d5x^ — x^ = 1556. 

1556 



In the first place, x is approximatively 



= 0,63, for which, how- 
0,4335 



3500 
ever, x = 0,65 should be put. From this we obtain 450 x^ = 450 
= 190,13, 35 x^ = 9,61, x* = 0,18, and finally 

_ 1556 + 190,12 - 7,56 + 0,18 _ 1738,7 _ 
^ ~ 2500 " ~ 2500 ~ 

and consequently the required thickness of metal is 

X 



0,695 inches. 



= 0,3475 inches. 



245. Excentric 

Fig. 389. 




2 

Load. — If the force which acts upon a 
girder supported at both ends 
A and B, Fig. 389, is not applied 
at the centre, but at some inter- 
mediate point, situated at the 
distances D A = l^ and D B ~ 
Zg from the points of support, the 
proof load is greater than when 
the force is applied in the mid- 
dle. Let us denote the forces, 
with which the points of support 
A and B react, by P, and P2 
and the entire length of the gir- 
der A B =^ k + Z2 by I. E"ow, 
if we put the moment of Pi in 



§245.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 481 

reference to the point of support B equal to that of P in reference 
to the same point and in Hke manner the moment of P^ in refer- 
ence to A equal to that of P or Pi I = P k and P^l = P Zi, we 
obtain the reactions at the points of support 

Pi = jP and P, = ^j P, 

and consequently their moments in reference to the points of 
application p 7 _ p ? _ ^ ^i ^2 

For any other point E, whose distance B E from the point of 
support ^ is = a;, we have this moment 

P,.B~E=?^- 
l 

smaller than that just found, and consequently at B we have the 
greatest deflection, and therefore we must determine the proof load 
in reference to this point alone, for which we have 
PI, k _ WT 
f ~ e ' 

moment of the force 



If we substitute ?. = - — a; and l^ = ~ + x, ^q obtain the 



Pkk 



^(|-')§-') ^S-") 



hence the proof load is 

p _ I WT IW T 



L\ ^2 ^ 



fi - '•) 



and therefore greater or less as x is greater or less. For x = -, 

z 

I.E., for ?i = 0, in which case P is transferred to the point of sup- 
port A^ we have ^ IW T 

-^=^77 = °°' 

and on the contrary for a; = 0, i.e. if the force P is applied at the 

centre, the proof load is a minimum and is 

W T 
P = 4: — — 

le' 

as we know already from § 240. A prismatical girder supported 
at both ends will sustain the smallest load, when the latter is ap- 
plied at the centre, and more and more as the weight approaches 
the points of support. 

If we lay off as ordinates the moments of the force, which are 



482 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 345. 



inversely proportional to the radius of curvature ^nd directly to the 
curvature itself, as ordinates at the different points upon the girder, 
we obtain a clear representation of the variation of the deflection 
at the different points upon the girder. 

PI L 

If, in the case just discussed, the moment of the force ^ 



I 



m 



D is represented by the ordinate D L and if from its extremity L 
the right lines L A and L B h^ drawn to the extremities of the 
abscissas D A = Ii and D B = l^, these lines will limit the differ- 
ent ordinates (as for example EN) representing the measures of 
the deflection for the different portions of the body; for since 
E N D L 



E B~ D B' 



EN 



it follows that 
E B 



D B 

as we had previously found. 
Fig. 390. 



D L^-^ 




Another case which 
often occurs in practice 
is, when the weight is 
equally distributed over 
a portion E F = c of 
the entire length I oi 
the girder A B, Fig. 
390. Let us again de- 
note the distances of 
the middle B of this 
weight from the points 
of support A and B by 
li and /s and the reac- 
tion of the abutments 
by Pi and P-^, then we 
have again 

and 



If Q were not distributed, but if, on the contrary, the force was ap- 
plied at P, the moment for D would be = — ^-, and, representing 



§245.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 480 

the same by an ordinate D X, the moment for the other points of 
A B will be cut off by the right lines L A and L B. But, since 
for the points within E F the forces Py and P^ act in opposition 
to the weight placed upon it, the ordinates between E G and F H 
will be diminished. For the centre D of the loaded portion E F 
the moment of half the weight 

must be subtracted, and there remains, therefore, of the ordinate 
D L = -^^i— ? only the portion 



DM 



DL~ML=Q(^-'l'-iy 



For another point N, whose abscissa is A iV, the moment is, on 
the contrary, 

We _ p {x-l, + ^ cf q 



P,.N A - NE .q. -^— ^ P, X - 



2 ' 2 

{x — l^ -\- ^ cY q 



and if Pi x is represented by the ordinate JV R and 



2 



by the portion S R, the ordinate JSf 8 will give the total moment 

r, X —^ . 

This is of course very different for different values of x, i.e. for dif- 

P\ 
ferent points, but is a maximum for x — l^ + h c = — ^, and then 

q 

its value is 

Hence we must put the proof load of this girder 
QIJ^ /-, _c_\__ WT 
I \ 21/ ~ e ' ■ 
Example. — What weight will a hollow parallelopipedical girder, made 
■of ^ inch thick sheet iron, support, if its exterior height is 16 inches and its 
exterior width is 4 inches, when it is loaded uniformly along 5 feet of its 
length, the middle of the loaded portion being 8 and 4 feet distant from 
the points of support ? Here we have 

ih' — b.h,^ 4.16^-3.15^ 

IT-^ = 16 = ^^^'^ 

and 



484 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 246. 



I 



(y-i-HM'-Mh 



32. 1 9 



76 
3' 



and the weigM required is therefore 

3 T 195,6 
Q = 391,2 • ^ • -g- = -1^ . 9000 = 23160 pounds. 

Remake. — If the weight Q is not uniformly distributed over JS F^ but 
if half is applied at the extremity E and half at the extremity F^ the line 
^ if ^ is then a right line, and the maximum moment is the ordinate 

G By for which 



Qh 



('. - 1) = 



WT 



l^ denoting the greater distance D A and l^ the smaller distance D B of 
the middle D from the two extremities A and B. 



Fig. 



t 246. Girders Fixed at Both Ends. — If a beam A B^ 
391, is loaded in the centre C smd fixed at hoth ends, it will be 

Fig. 391. 



>^ 



V 



^ 




curved upwards at the centre, and at the points of support A and 
B downwards, and there will be formed at the centres D and E of 
the semi-girders C A and C B points of inflection, where there is 
no curvature or where the radius of curvature is infinitely great. 
One-half of the weight P is supported hj A D and the other half 
by B E, and we can therefore assume that both the quarters A D 
and B Eoi the beam are bent downwards at their ends D and E by 

p 

— , and that, on the contrary, the half D E oi the girder is bent 

upwards at its ends D and -£' by /— — ). The arm of each of these 



§246.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 485 



forces A D = CD, etc., is 



A B I .1 X. . 

— — = -; consequently their mo- 



ment is 



PI PI 

-— . - = — -, and therefore 
2 4 8 



PJ 
S' "^ 

P = 



_ W_T 
e 



; hence we can put the proof load 



^ 2. 



A:WT 



I e - " • I ^ 
Such a girder will bear twice as great a load as when it is 
simply supported at both ends. 

PI 



C L = -^, and 

o 



If we make the ordinates A H = B K 

draw the right lines H L and K L, they will cut off ordinates 

{M N) for every other point (Jf ) upon the beam proportional to 
the moments of the force and to the deflection. 

If in the formula, which we have found, we substitute the modu- 
lus of rupture K instead of the modulus ot proof strength T, we 
obtain, of course, the force necessary to break the beam., which is 

8 WK 



P = 



I e 



Since the curvature is the same in ^, ^ and C, the rupture will 
take place at the same time in A, B and C. 

If the position of the girder is the same and the load Q := I q 
is uniformly distributed, the girder assumes, it is true, two curva- 
tures upwards and two downwards, but the points of inflection 

Fig. 392. 
-Ji -R 




D and B, Fig. 392, do not he at the centres of the semi-girders ; 
for the deflecting forces R, R of the portions A D and B E are 



486 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 246. 



aided by the weight upon the latter, and, on. the contrary, the 
action of the bending forces ~ R, — R of the central piece D is 
diminished by this load. Let us put the length A D = B E = l^, 
the length C D = C E — l^ and the total length of the beam I = 
^ (^1 + h), and let us denote the weight upon A E or B E hj 
Qi = q li, and that upon E E hj Q^ =^ 2 R — 2 q l^. l^ow, since 
^ Z> is bent downwards by R and ft, we have, according to § 216 
and § 223, the angle of inclination to the horizon E E T — E E T 
= a at the point of inflection E 



a 



+ 



2 WE ' 6 WE' 
and since C E is bent upwards by (— R) and downwards bT §, 
we have for the same position E also 



25 



a 



2 WE 6 WE 

Equating the two values of a, we obtain the relation 
3 R ik' - I,') = ft // + Q, l,\ or 
^qk {h' - I') = q {l. + I2'), I.E., 

3 ?{?.^ - g - ^.j; 

Eesolving this equation, we obtain 



lo' 



6 - '.)■ 



U = iV'^ Midi, = i (1 



^'i), 



and, therefore, the moment of force in relation to the middle G is 

Rk _ Rh _ qU _ qj_ _ 0} 

" ~ 24' 



M = RL 



2 2 2 24 

and that in reference to the extremity A or B is 

Fig. 393. 



-E 



-E 




§247.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 487 

= ^' (1 - iTj) (1 + VT) 

- g ^' (1 - i) = ^^ = o il 
~ 8 12 24' 

The proof load of this beam is therefore 

/i_ io WT _ 3 8 WT 

I.E., I times as great as in the former case, where the weight acted 
at the centre C. i 

If we lay off -ttt as ordinate in A and B and also ^— - as ordi- 
•^ 12 24 

nate in O, making A H = B K = -—- and L -- — -—--, we ob- 

tain three points H, K and L of the curve H D L E K, which 
represents the variation of the deflection of the girder. 

Example. — How high can grain be piled in a grain house, when the 

floor rests on beams 25 feet long, 10 inches wide ana 12 inches high, if the 

distance between two beams is = 3 feet and if a cubic foot of corn weighs 

46,7 pounds 'i If we employ the last formula Ql = VZ . 167 . 5 ^^, we 

must put 

5 = 10, ^ = 12, Z = 25 . 12 = 300, and consequently 

12 . 167 . 10 . 144 
Q, = wqq = 9619 pounds. 

Now a parallelopipedical mass of grain 25 feet long, 3 feet wide and 
X feet high weighs 25 . S . x . 46,7 pounds ; if we substitute this value for 
§, we obtain the required height of the mass 

9619 
^ = 75T1677 = ^'^^ ^''*- 

§ 247. Beams Dissimilarly Supported. — If a beam ABO, 
Fig. 394, is fixed at one end A and supported at the other B and if 
the load acts in the middle between A and B, we have, according 
to § 221,, the reaction of the support B 

P - ~ P- 

and therefore the moment of the force in reference to 

PJ 5 



^ ^- 2 -3~2^^' 
and, on the contrary, that in reference to A is 



488 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 347. 



^^ ^l 



AH= P 



2 



P^l = Fl[^ 



\2 16/ 



— PI- — PI 
16 ^ ^ ~ 32 ^ ^^ 



Fig. 394. 




or greater, and consequent- 
ly we can put the proof 
load 



P = 



16 
3 



WT 

le ' 



For an intermediate point 
M, at a distance C M — x 
from the centre G, this mo- 
ment is 

MW= P, (^ + ^ 



- P X- P^ 



I 



If we assume x = 



1 P 



I 



2 

- (P - PO X. 

1 5 
- — —~ L we obtain 

2 22 



P - Pi 16-5 
that point, for which the moment is equal to zero and the radius 
of curvature infinitely great. The variation of this moment and 
the deflection of the girder are represented by the ordinates of the 

right lines II L and L P, passing through the extremities of A II 

= 4 ^ ^ and of (TL = ^Pl 

If, finally, a girder A B, Fig. 395, supported in the same man- 
ner as the last, is uniformly 
loaded, as we have previous- 
ly generally supposed, with 
a certain weight q upon the 
running foot of the girder, 
we can determine the reac- 
tion Pi at the support B in 

If 



Fig. 395. 




the following manner. 



the length of the beam is ?, 
the entire load is Q = I g 
and the moment of the force? 
in reference to a point i/, 
at a distance B M = x from 
the point of support P, is 



^ 



}' 



§248.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 489 

and consequently the angle of inclination 

^ - 2 WE 6 WE ' 

and (according to § 217 and § 223) the corresponding deflection is 

^ ~ 2 PF^ 6 r^ • 

But since A lies on the same level with B, the ordinate in A, 
I.E. for X = 1,1^ y ^= 0, and we must put 

from which we obtain the reaction at B 

-^1 ^^ 8 3' ^ ^^ 8 ^* 
If we substitute this value for Pj in the expression for the mo- 
ment, we obtain 

R S = ^ Q X — -— = ^ (^ I — x); and therefore foYX = l 

_ ql" _ Ql 



AH 



8 8 

For X = B D = I I this moment is = 0, and for x =. B E =. 
I Z it is a maximum 

_ 9 ^ r _ 9 



^^=12^-128^^- 



Ql 16 ^ , 9 



Since -^ = -— Q I > --^ 6 ^? the moment ^ jff in reference 



to the fixed point A is greater than the moment K E in reference 
to the middle E oi B D, and the proof load corresponding to the 

moment ^ must therefore be determined, i.e. we must put 

W T 

in which case we assume, of course, that the modulus of proof 
strength for extension is the same as that for compression. 

This proof load is 8 . -y^^^ = :^ times as great as it would be if 
the weight were concentrated in the middle. 

§ 248. Girders Loaded at Intermediate Points. — If a 

girder A B, Fig. 396, loaded at both ends with equal weights P, P, 



490 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 248 



Fig. 396. 




is supported at two points C 
and D, which are at the same 
distance A C= B D =1^ from 
the ends, the reaction of each 
of these points of support is 
equal to the force P, and for a 
point if upon CD the moment 
of flexure 



C L = D = MN 

= P {x,- I,) -Px,- -PI, 

is constant, and the form of 

neutral axis of C D is therefore a circle, while, on the contrary, for 

a point U upon A G this moment U V —- P x \^ variaUe and 

smaller than P li. 

WE 
The radius of curvature of the middle piece C i) is = r = -p-^- , 

and the angle of inclination of the axis of the beam in C and D is 

I denoting the length of this 



I Pllx 

consequently «i = ^- = ^^^r^, 

middle piece. From this we obtain the deflection 

~ 8r 



M8= a = 



2r 



Cti =: «! Z, + 



Pl^' 



8 WJE' 
PI I,' 



as well as the deflection of C A 



+ 



Plr' 



Pin I 



3 WU 2 WB ' 3 WIJ 



WE 



2 3/ 
W T 



( - )• 



The moment of proof load for this girder is P /i = — 

If the same beam A B w> uniformly loaded, as is shown in Fig. 

397, with q per running 
Fig. 397. f^^^^ under certain circum- 

stances the moment of 
flexure for some points is 
i^ positive, and for others 
negative, and therefore at 
two points U and V it is 
equal to zero. 

For a point upon A C 
and B D this moment is 
^ q x", and, on the con- 
trary, for a point between 
C and the middle M, or between D and M, since the value of the 

reaction at C and Z) is ^ Q = {^ I + l^) q,\t\s R 8 = y =^ I 




§ 249] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 



491 



{x + l\Y q — {^2 I -\- l^) X q = ^ {x" — I X + l^i) q, and therefore 
— for o;''^ — I X =^ — l^', i.e. for 



C U= X 



li^ and for 



I 



C V= X =-^ + 
which of course requires that l^ 



•''&) 



7^ 



2^ 



< -, I.E. C A<:GM. Under 



Fig, 398. 



any other circumstances the moment of flexure remains always 

positive, as is shown in Fig. 398. 

The moment of flexure is a maxi- 
mum or minimum for x = - and 

while the moment of flexure in C 




andDis C L = D = iq l,\ 
If, therefore, in the first case, 

Fig. 397, g- ?,»>?.' or g> 
2 V, i-E. l> kV^, we have M IST 



Q 



> C L, and since q = ^ — ^yy? we must put the moment of proof 
load equal to 



2(Z+ 2 I,) 



while, on the contrary, we haye 



Ql.' 



W T 



, when Z < Zi '/S. 



>u 



Fig. 399. 



2(/ + 2/i) e 

§ 249. Grirders not Uniformly Loaded.— If a beam A B, 
Fig. 399, is not uniformly loaded, but in such a manner that the 

load on the running foot increases 
towards the extremities of the girder 
regularly with the distance from its 
centre, the statical relations will be as 
follows. 

If / == ^ ^ = 2 C A = ^ CBis 
the length of the beam, measured be- 
tween the points of support A and B, 
q the weight of the load per unit of 
surface of the cross-section and p the 
angle of inclination A C D —B C E 




492 GENERAL PRINCIPLES OF MECHANICS. [§250. 

of the planes O D and C E, which bound the load, we have the 
weight of the prism A G D — B C E of the load, sustained by one 
point of support, 

^ = J A~C, AD , q = ^ {^^ tang, p . q = ^^ q r tang, p, 

and consequently the moment of this force in reference to a point 
iV, at a distance A N = x from A, is 

y\ = -^'X = lqrx tang. p. 

The weight of the heavy prism above A N=x\^ q I — — 1 A N, 

and the centre of gravity of the same is at a distance N ~ 

. ^ T,^ -r . —p— from iV", and consequently the moment of 

AD -h N L 3 ' ^ -^ 

this prism in reference to iV" is 

i/i = q (2 A D + JV L) — - — = q\l tang. P + (o "" ^) ^^^9' P ~a 

o x^ 
= ^ tang, p (i I — x), 

and the entire moment of flexure for the girder at iV"is 

WU^y = y,-y,= ll^^tl (3 ^a; _ 6 Ix' + 4.x') 

- = ^-^^{^l^-^lx^^x^)=.l[i^^ 

if we put C N = Xi = - — x OY measure the abscissa Xi from C. 

I a r 

This is a maximum iov x = - and equal to -— tang, p, and 

the moment of proof load of this girder is 

q P , Ql WT 

^^ tang, p, i.e., -^- = —^, 

while for an uniformly loaded beam the moment of flexure is 

Q I W T 
hence the moment of proof load is ~- = . 

8 e 

' § 250. Girders Sustaining Two Loads.— If a girder A B, 

Fig. 400, supported at both ends is loaded at a point C, which is at 
the distances C A — ly and C B — l^ from the j)oints of support 



n 



i' 



§250.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 493 



A and B, with a weight P and in addition carries a uniformly dis- 
tributed load Q = q I, the reaction of points of support A and B 

IPO I P Q 

are Ri — —, h -^ and B^ = --S~ + o ' ^^^ ^^® moment of 

flexure at a point iV", situated at a distance J. iV^ = ic from the 
point of support A, is 



Fig. 400. 



Fig. 401. 





This moment is a maximum for 
2 J?i 



Br 



— X = X, I.B., for a; = • — , and is then 

q . ' q 

It is here assumed, that C A > C B, i.e., ^i > 4 and x < ^j. 
If a; ^ ^1 the maximum of the moment of flexure is at C (Fig. 401), 
and consequently 

y-CK-HJ,-—- — P +~ —- _yp + -J —. 

If we substitute 

Bi (kP Q\ I 
= - = (— + |-)^ = A,weobtam 

and the moment of;proof load of the girder, when 

L < L:iA is 



a; 



494 



GENERAL PRINCIPLES OF MECHANICS. 



[§251 



( — -^ + -~-l — yr = , and, on the contrary, when 



Q 



2 4 



Q 

it is 



{'+!) 



Q\lih WT 



Fig. 402. 





111'' . i -ii 


uuutiiHIIIUIIIIIIIIIIIIIIIIIIIIIIIIMIIII[|i;iliill!llill{1l{lllilllllll| 


c 


i'lH-' '|.'l,il 


W'r\ ■ i'liil 


.!''IHVVl.-:|Wl.l,i|ilin| 



B 



2/ I e 

These formulas are specially applicable to cases, where the 
weight of the beam is taken into consideration ; here G must 
be substituted for Q. 

.-J 
§ 251. Cross-section of Rupture. — In all the cases, which 

we have previously treated, we have assumed the body A By 

Fig. 402, to be prismatical and, there- 
fore, the moment of flexure W E to 
be constant, hence we could conclude 
from the fundamental formula 

P xr = WE, 
that the radius of curvature • 
WE 
Px 
was inversely, or the curvature itself directly, proportional to the 
moment (P x) of the force P acting upon the body and that con- 
sequently the curvature becomes a maximum or a minimum at the 
same time that P x does. 11^ therefore, the force P is constant, 
or if it increases with x (as, e.g., in the case represented in Fig. 403, 

where Q =z q x), the curvature in- 
creases or diminishes with x and be- 
comes with it a maximum and mini- 
)ii|i |:!i;i|t!|ip'si!miiiiif^^ mum. When, on the contrary, the 

cross-section F of the body is differ- 
ent in different points, then W = 
2 (F z^) is also variable, the radius of 
curvature is proportional to the quo- 

W 

tient -^^— and the curvature itself to 
Px 

P X 

the expression -^. If we are required to find the points of great- 
est and least curvature, we have only to determine those, for which 



Fig. 403. 




Px . . 

-zf^ is a maximum and a minimum. 
W 



§ 251.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 495 

In like manner, according to the formula 

^ P xe 

'^ = -r 

of § 235, the strain iS^ in a hody is proportional to the expression 
, and becomes a maximum or a minimum simultaneously 

with it. 

W 
If the body is prismatical, — is constant, and the maximum 

strain S is proportional to the moment P x of the force only. If 

W. 

the cross-section of the body varies, — is a variable quantity, and 

e 

this strain is dependent upon this quotient also. In the first case 
the strain becomes a maximum with P x, e.g., when the beam is 
acted upon at one point by a force P and by a load Q — q x uni- 
formly distributed over a distance x, for x = I; in the second case 
this maximum cannot be determined unless we know how the 
cross-section varies. In order to find the point of maximum strain, 
it is necessary to determine by algebra the maximum of the expres- 

P X e 
sion J^-, , In any case the part of the body where this maximum 

strain occurs is also that point at which, if the load is sufficient, the 
strain /S' first becomes equal to Tand also to K, and, consequently, 
where the limit of elasticity will first be attained or where rupture 
will take place. This cross-section of the body corresponding to 

the maximum value of ( ) is therefore called the section of rup- 

ture (Fr. section de rupture, Ger. Brechungsquerschnitt) or also 
the dangerous (weak) section. 

If the body has a rectangular cross-section, with the variable 
width u and the variable height v, we have 

W _ uv' 
e ~'6 ' 

P X 
and the section of rupture is determined by the maximum of 



u V 



u V 

3 



or by the minimum of -^ — . 
•^ P X 

For a body with an elliptical cross-section, whose variable semi- 
axes are u and v, we have 

W rr u v'^ 



9 



496 



GENERAL PRINCIPLES OP MEOHANICS. 



[§ 352. 



and we must therefore again determine the minimum value of 



u V' 



p -, when we wish to know the weakest point in the body. 

When the weight is constant, F can be left out of consideration, 



uv 



and we have to determine only the minimum of — . If, on the 



X 



contrary, the weight Q = q x is uniformly distributed upon the 
girder, we must determine the minimum of — j- in order to find the 
section of rupture. 



Fig. 404. 



§ 252. If a body A CDF, Fig. 404, forms a truncated wedge 
or a horizontal prism with a trapezoidal base A E B F, whose con- 
stant width i^ B C = D E = 1), and if the force P acts at the ex- 
tremity D F oi the same, we 
have to find only the mini- 

mum of - m order to deter- 

X 

mine the section of rupture. 
Putting the height i> 6^ = 
E F oi the end = h and the 
height K U oi the truncated 
portion H K U = c, and as- 
suming, as previously, that 
the section of rupture L M N is at a distance U V = x from the 
extremity D E F,we obtain the height of this section 




ML = V = h -\-^h 

c 



(' * ')■ 



and we have therefore but to determine the minimum of the ex- 
pression 



X X \ c/ \x c c / 



1 X 

or, since h and c are determined, only that of — I — ^ 

X 



2 



If we assume x = c, the latter expression becomes = -; but if 

c 



we make x a little {x^) greater or less than c, we obtain 
1 _ 1 _ 



(l ±:^) ^ ^ ^ ^^ 



and 



§252.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 497 



consequently 



c r C 



1 X _% x^ 



hx — ^ = Ih 



2 
or in any case greater than -. Hence x = c gives the minimum 



required, i.E. the section of rupture L M JV is at a distance from 
the end D E F equal to the height E U = c or to the distance of 
the truncated edge H K from the same end D E F in the other 
direction. 

The height of this section of rupture is 

c 
and consequently the proof load is 

p_ h (2 ny T_ __ 4 ^ h' T 

~ c • 6 ~ c • 6' 
For a parallelopipedical girder, which has the same length ? = c, 
the same width h and equal Yolume V = hhl, the height is 

h ■\-_%li 
2 
and consequently the proof load is 

~ c • 6 ~ 4 c • 6' 
and such a girder bears, therefore, but /g as much as the wedged- 
shape body just treated. If the body is a truncated pyramid, the 
edges A E, B D, etc., when sufficiently prolonged, cut each other 
in a point, and if we designate the height of the truncated portion 
by c, we have 

MN= u = h{l + |)andXif=v = a[i + -]• 

and therefore the minimum of 

X ~~ X \ c ) 

or of 

1 Bx x^ 

X c c 

must be determined, in order to find the section of rupture. By 
the differential calculus we obtain 

x = ic 
32 



498 GENERAL PRINCIPLES OF MECHANICS. [§353. 

and we can easily satisfy ourselves that this value is correct by first 

substituting x — ^ c + x^ and then x =z i c — x^. In both cases 

we obtain a greater value than 

2 3 1 15 

- + 7^ — \- T- = :i-, which is the value 

c 2c 4c 4c 

the expression 

— I ^ H r, assumes for a? = i c. 

The distance of the section of rupture from the end D F is, then 
equal to half the height c of the portion of the pyramid, which is 
cut off. The dimensions of this surface are 

and, consequently, the required proof load of the beam is # 



c 6 4 c 6 



For a body, the form of which is a truncated cone, we have, 
when the radius of extremity is r and the height of the truncated 
portion is c, the radius of the section of rupture Vi = ^ r, and 
therefore 

^~T • c • 4* 

V. § 253. Bodies of Uniform Strength.— If a body is so bent, 
that the maximum strain 8 upon the extended and compressed 
side of the neutral axis is at all points the same, we have a 'body of 
the strongest form, or of uniform strength (Fr. corps d'egale resist- 
ance, Ger. Korper von gieichem Widerstande). By a certain force 
such a body is strained to the limit of elasticity in all its cross- 
section at the same time, and has, therefore, in each part a 
cross-section corresponding to its proof strength ; it requires, 
therefore, when the other circumstances are the same, a smaller 
quantity of material than any other body of the same strength. 
Therefore, for the sake of economy and to avoid unnecessary 
weight, such forms are to be preferred in construction. 

Since the greatest strain in a cross-section is determined by 
the expression 

-S = ^ (see i 351), 

P X e 
a body of uniform strength requires that ^ shall be constant for 

all cross-sections of the body. 



253.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 



499 



If the force P is constant and applied at the end of the body, 

we have only to make 

e X W 

-:= or ■ 

W ex 

constant, and when the force Q = q x i^ uniformly distributed 

upon the girder. 

e x' W 

-^ or — - 

W ex 

must be constant. For a girder with a rectangular cross-section (see 
§ 251), whose dimensions are u and v, we must make in the first 

case , and in the second — - constant. 

If at another place at the distance I from the extremity the 
width is h and the height li, we must haye consequently in the 

111' 

For the constant width u = h, we have in the first case 

1^ ¥ 



n . U V hit T . ,1 ^ U V 

nrst case = -7-, and m the second — - 

X I x' 



X 



I 



-, I.E., 



V 



X . 



X 



Since the equation — — -- is that of a parabola (see § 35, Ee- 
mark), the longitudinal profile A B E, Fig. 405, of such a body 
Fig. 405. Fig. 406. 





has the form of a parabola, whose vertex E coincides with the ex- 
tremity or point of application of the load P. 

If a beam A B, Fig. 406, lohose loidth is constant, is supported 
at both ends and sustains the load P in the middle, or if the beam 



500 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 253. 



Fig. 407. 

A2P 



A B, Fig. 407, is supported in the middle and is acted upon at its 
ends A and B by two forces, which balance each other, its eleva- 
tion must have the form of two para- 
bolas united in the middle. As ex- 
amples of the latter case, we may 
mention working beams, balance 
beams, etc. As the beam is weak- 
ened by the eyes, made for the shafts 
A, B and (7, lateral or central ribs 
are added to it. 

If the height v ^= h \^ constant^ 
we have 

It _ X 

l~V 

and the width is proportional to the distance from the end ; the 
horizontal projection of the beam AGE, Fig. 408, is a triangle 
BCD and the entire girder is a wedge, the vertical edge of which 
coincides with the direction of the force. 




u 

X 



- = V or 



Fm. 408. 



Fm. 409. 





Instead of the parabolic girders. Fig. 405, we generally make 
use of girders, Fig. 409, with plane surfaces. In order to econo- 
mize as much material as possible the girder is made in the mid- 
dle M of the same height M = h„, = h V ^, as the parabolic 
girder would have been, and the limiting plane surface CD is made 
tangent to the corresponding parabolic surface. We have 
B C _ 3 AM _ 3 A^ _ AM _ ^ 
MO~2AM~-MO~2AM~^ 
and consequently, if we denote the greater height B C hj hi and 
the lesser one A Dhj As, we obtain 

hi = ih^==ihV~J= 1,0607 h and 
h, = hh^ = ihVJ= 0,3536 h, 



§ 254.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 501 



for which we must determine the height B JSF = hhy means of 

the well-known formula P I = b h' -^. 

The volume of such a girder, whose faces are planes, is 
bljhi + h,) ^ 0^^071 b I h, while that of the parabolic girder of 

equal strength is = | ^> ^ ^ = 0,667 b I h, i.e., 5,7 per cent, smaller. 

In like manner we can 
construct the girder A NA^, 
Fig. 410, which is supported 
at its extremities A and^i, 
of two portions, bounded by 
plane surfaces, which have a 
common height B C — hx =^ 
1,0607 h at the point of ap- 



FiG. 410. 




plication of the load, and at the extremities the altitude 

A D = A,D, ^ h, = 0,3536 h. 
Here the altitude B N — li must be determined by the formula 

PI, h __ b ¥ T 
'-^.v ^ ~ 6 

§ 254. If the body A B D, Fig. 411, is to be made with all its 
cross-sections L M N, A B C, etc., sifnilar, we must put 



Fig. 411. 



T = T and therefore 
h b 




u . tc"" h^ 



b'x 



I.E., 



U^ 

T 



X 



bh^ 

u 



V ./x 



The width and height are therefore 
proportional to the cube root of corres- 
ponding arms of the lever. When the 

distance fro^l the end becomes eight-fold, the height and width 

are only doubled. 

We can replace this body by a truncated pyramid A C E G, 

Fig. 412, at the middle of whose length the height is i¥ = h„, == 

'^l.h = 0,7937 h and the width M N =^ br, = '-^J, .b = 0,7937 b 
and the strength of this body is exactly tlie same as that of the body 

just discussed. For the tangential angle of the curve - = y , or 
V — ^~ x\ we have, according to Art. 10 of the Introduction 



502 



GENERAL PRINCIPLES OF MECHANICS. 



[§354. 



Ji 



to the Calculus, tang, a = o~n= 
that for 



X~s = 



h 



3 V ix' 



r, therefore it follows, 



= hi ^ tf^'T^g- « = '^^^\x) 



X 

1 

and in like manner we have for the curve 



^ y^ V4 = I V^ = 0,2646 h, 



u 



= yi 



J, tang. /3 = 3/-—^ and 



^ I tang, i^ = is ^i- 



Erom this we can calculate the dimensions of the base ABO 
A B = h, = kn + i i tang, a = aV^ ,h = 1,0583 h and 
B C =bi = i^ -}■ il tang. /3 = | ^T . j = i^0583 b, 
and those of the smaller base E F 

Fig. 412. Fig. 413. 

C B 





F G = h, = hrr,- ^l tang, a = f V^ . /^ ^ 0,5291 h and 
E F = h = K- ^l tang. /3 = f v/r , j ^ 0,5291 h 

hli" T 
We must of course put P I =^ — - — . 

If we make the cross-section of the body of uniform strength 
circular, we have for the variable radius the equation 



u 



V 



X 

V 



and if we replace this body by a truncated cone ABE, Fig. 413, 
its radii must be 

Jf = 7"^ = v^i . r =r 0,7937 r, C A = r, = 1,0583 r and 
D E = r, = 0,5291 r, 
and the radius r of the base of the solid of uniform strength must 
be calculated according to the formula 



^.254.] ELASTICITY AND STRENGTH OF FLEXUEE, ETC. 



503 



If the girder is uniformly loaded and its width is constant^ i.e. 



iiu = h, we liaye 






V _ X 

7i~V 

and its form must be "that of a wedge, whose elevation is a trian- 
gle A B D, Fig. 414. 

Fig. 414 Fig. 415. 





If the height is constant, we have r — -w\ hence the horizontal 

^ hi 

section of the girder is a surface limited by the two inverted arcs 
of a parabola B D and C D, as is shown in Fig, 415. 

1t^ v'^ '3^ 

If we affain make the cross-sections similar, we have -t-^= t^^^, 

and the vertical and horizontal profiles are cubic x>arabolas, the 
cubes of the ordinates of which are proportional to the squares of 
the abscissas. 

If a body ^ E B, Fig. 416, supported at both ends, is uni- 
formly loaded with the weight q 



Fig. 416. 




iiiiiilniHi 
iiiimiiiiil i: 



iMJj 



2 '2 



per running foot or upon its whole 
length A B — I with Q — q I, we 
have the moment of the force at 
a point 0, situated at the distance 
A = X from one of the sup- 
ports A, 

X q /- 2\ 

~ q X , ~ — ~ yt X X j, 

and, on the contrary, at the cen- 
tre C 

Q l __ Ql __ qj^ 
2" "4" ""8 " 8 • 



2 *'' 



Assuming the width Z» of the body to be constant, we have 



504: 



GENERAL PRINCIPLES OF MECHANICS. 



[§355. 



Iv' 



T q 



= j-{J,x — x^) and 



6 2 



^^ • 6 ~ 8 ' 
Ji denoting the height G E oi the body at the centre, and by divi- 
sion we obtain 

V^ I X — x^ 

-, or 



¥ 



1 p 

4 ^ 



v' = {~)\lx-x^y 



lih — II, v"" would \)Q — I X — x"", and therefore the longitu- 
dinal profile would be the circle A D^ B, described with the 

radius ^l', but since I x — x"" must be multiplied by ( p-J in order 

to obtain the square v"" of the height if = iV at any point, the 
circle becomes an ellipse A D B or A U B, whose semi-axes are 
CA = a, = U and CI)=GF=I}, = h. 

We can replace this body by a girder A A B D B, Fig. 417, 

with plane surfaces, whose 
height at the distance A M 
K . ■ — gf^ = \ I from the points of sup- 

port B and B is M = h^ 

The angle of inclination a of 
the surface B D to the axis ^ Cis given by the equation 

h i^ - ^ 2 h 



Fig. 417. 




tang, « = t] • 4/"/ 



il 



X 



X 



I ' Vj% r 1V3 



2 h ,^ h 

— 7^/0—3 ^ ^ • 7 ? 



I 



consequently we have ^ tang, a =z J V3 . /^ and the height of the 
body in the middle 

GD= MO + ^-r tang. a = iVS.h = 1,1548 h, 
and, on the contrary, the height at the ends is 

AB = MO-j tang, a = i VS . Ji = 0,5774 h. 

(§ 255.) The deflection of a body of uniform strength is, of 
course, under the same circumstances, greater than that of a pris- 
matical girder. For the case, where the beam is fixed at one end 



I3255.] ELASTICITY AJ^D STRENGTH OF FLEXURE, ETC. 505 

and subjected to a stress P at the other, the deflection is found as 

r E 
follows. The well-known proportion - = -~ gives us the formula 

& JL 

E e 
r = —FfTi ^^ which the radius of curvature is a function of the dis- 
tance e. If we know the dependence of e and x upon each other, 
we obtain an equation between r and x, from which we can deduce 
(in the way explained in § 218) the equation of the co-ordinates of 
the elastic curve. If we assume the deflection to be small, we can 
again put the length of arc s equal to the abscissa x, and conse- 
quently equate the differentials d s and d x ; hence we can, as be- 
fore, assume d x 

r = ~ -J-, 
a a 

From this we obtain 

dx=— ^eda, 
and, by integration, the tangential angle 



T rdx 



E'J e' 

For a girder with a rectangular cross-section e = ^ v, and 

therefore % T rd x 

E ^ V 
If the width is constant or u = b, we have 



v^ X 



— = - (see § 253), and therefore 
y = Ji y —. and 

V 



2 T Vl r . . 2 T Vl 

a = — 



. -J xr-\ dx = =- . -J- . 2Vx + Cons,f 

or, since for x = I, a = and consequently 

n 2T VI 

If we put a = -^, we obtain 
d X 



, AT Vl ^ 

«y = -^ -j^iVl - V^)d 



vC/« 



and, therefore, the required equation of the co-ordinates is 
4 T VI ^_ ,_ T VI ,_ 



506 GENERAL PRINCIPLES OF MECHANICS. [§256. 

For X = l,y becomes «; the deflection is then 

T P 



""-'Ell 



T 6 F I 

But PI = bh'' .-or T= -^^, and, therefore, the deflection 

is given by the formula 

^ Ebh'~ ^' Ebh'' 

I.E , it is twice as great as in the case of a parallelopipedical girder, 

whose height is h and whose width is h (compare § 227). 

If the force acts at the middle of a girder, supported at both 

P 1 

ends, we have only to substitute — for P, and - fpr I, and we obtain 






2 ' 2 

s pr 



'^ ' Ellf 

LE., it is 16 times smaller than when the force acts at the end. 
For a body of uniform strength with a triangular base, as is 

represented in Fig. 408, the variable width is u = j b, and 

„ II h^ T^ bit X ^ 

hence the radius of curvature r = -^ . -p is constant, the curve 
formed is a circle, and the corresponding deflection is 

- _L - ?_:?_? - 3 ^^ ^ 
^ ~ 2r~ bh'E~ ^' bh'E' 

I.E., I times as great as for a parallelopipedical girder. 

§ 256. Deflection of Metal Springs. — The most common 
examples of bodies of uniform strength, as well as of those which 
bend in a circle, are steel or other metal springs. The springs, of 
which the spring dynamometers are made, are of the finest steel and 
are from A to 1 meter long, from 4 to 5 centimeters wide and in 
the middle from 8 to 21 millemeters thick. They form bodies of 
uniform strength, and their longitudinal profile is composed of two 
parabolas united in the middle (see § 253). In order to increase 
the action, the spring dynamometer is made of two such parabolic 
springs A A and B B, Fig. 418, which are united at their ends A 



§ 250.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 507 

by means of the links A B,A B (see Morin's Le9ons de Mecanique 
■pj^ ^-^g Pratique, Resistance des Materiaux, 

TNo. 198). These dynamometers 
A measure the force P, which is ap- 
B plied to the hook D in the middle 
of one of the springs, by the space 
« described by the point Z, which is 
of course equal to the sum of the 
deflections of the two springs. But 
from what precedes we know that 

_ _^ 8 Pf 

and consequently we have here 

s ~ 2 a — 



h ¥ E' 
and, therefore, the force 

corresponding to the space s described by the pointer. 

In experimenting with such an instrument, whose springs were 
of the following dimensions: h = 0,05, h = 0,0211, 1 = 1,0 meter, 
the space described by the pointer was s = 9,1! millemeter, when 
the load was F — 1000 kilograms ; the coefficient of this dynam- 
eter was therefore 

**1Z _ Z - 1000 - 103 09 

and for other cases we must put 

P =. 103,09 s kilograms, 
when s is given in millimeters, or when the scale is divided into 
millimeters. 

If, instead of parabolic springs, we employ triangular ones of 
uniform strength, we have 

~ = a = -r^T. . -, ^^ -ry , and, therefore, 

p = i('4-)-. 

I.E., one-third greater than for a dynamometer with parabolic 
springs. 

Wagon springs should unite great flexibility with great strength, 
Avhile, on the contrary, it is not necessary to know the exact relation 
between P and s. For this reason, these springs are often formed 
of a number of simple springs laid upon one another. 



508 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 256. 



If the compound spring is composed of n simple parallelopiped- 

ical S2)rings, placed upon one another, we have, when the width is 

Z>, the thickness h and the length I, the deflection corresponding to 

4 P f 
the force P at the end A of the entire spring a = 



nEhh' 



and the 



proof load 



th^ T 
P — n —^ — ttj and*fcherefore also 



T f 



a 



T I 



^~^^ E n"^^ l~''^ E K 



If the entire spring A C D, Fig. 419, consists of n simple tri- 
angular springs, we have 

QPl' ,., ^ h¥ T 

a = — YTT-i-zy while P = n -^~ -^ 
n El)¥ I ^ 

remains unchanged, and therefore 

_T V a_T I 

'^- E Ji''^ l~ E }i 



a 



Therefore, in both cases the measure j of the flexibility in- 

T I 

creases with the ratios ^^ and t and is the same as for a simple 

spring of n times the width {n 5). 

Fig. 419. Fig. 420. 





In order to economize material, we superpose springs of differ- 
ent lengths and construct them of such a shape, that by the action 
of the force P at the end A of the entire spring they are bent in 
arcs of circles of nearly or exactly the same radius. The force P 
bends the lowest triangular piece A A of the the entire spring 

ABU, Fig. 420, whose length = -, in the arc of a circle, whose 

radius is r = n — ^ . -^, and in order that the remaining paral- 
lelopipedical portion shall be bent in like manner, it is necessary 



^256.] ELASTICITY AND STRENGTH OF FLEXURE, ETC. 509 

that the same shall exert a pressure at A upon the succeeding 

spring, which shall be equal to the force F ; for the moment of 

P I 
flexure of this spring is then equal to the moment — of a couple 

71' 
7 

(P, — P) whose arm is -. The relations of the flexure of the first 
spring repeat themselves in the second, which is - shorter than 

it ; it is bent in a circle whose radius r = -^r^j- • 75-, when its end 

A I ^2 is triangular and the other portion is parallelopipedical, and 
if it presses on the third spring with a force P. This is also the 
case for the third spring A^ G B, etc., up to the last piece, which has 
no parallelopipedical portion, and which, by the action of the force 
P, is bent in a circle of the above radius r. The entire deflection of 

r 6 p r 

this compound spring is ^^ = — = — , and the proof load is 

P = n —^~ -/r, hence 

^~ E h'^^ I ~ E h' 

The relations of the flexure are here exactly the same as for a 
spring composed of single triangular springs; it can also easily 
be proved, that both sets of springs require the same amount of 
material. 

It is not, however, necessary to make the ends of the springs 
exactly triangular ; we can employ any other form of equal curva- 
ture, E.G., we can make them of the constant width h and then at 
the distance x from the end A the height must be 

, . /nx 

y = hy j- 

Such a double spring is represented in Fig. 421. Here the 




total proof load is 2 P ; the length must not, however, be meas- 
ured from the middle, but from the ends B D, B D of the fastening. 



510 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 257. 



Remakk, — The reader can consult upon the subject of wagon springs : 
F. Reuleux : Die Construction und Berechnung der fiir den Maschinenbau 
wichtigsten Federarten. Winterthur, 1857 ; also Redtenbacher : die 
Gesetze des Locomotivenbaues, Mannheim 1855, and Philips : Memoire sur 
les ressorts en acier, etc., in the Annales des Mines, Tome L, 1853. 



CHAPTER III. 



THE ACTION OF THE SHEARING ELASTICITY IN THE BENDING 

AND TWISTiNG OF BODIES. 



§ 257. The Shearing Force Parallel to the Neutral 
Axis. — In a body, which is subjected only to a tensile or com- 
pressive force, the bases A B and C D of an element A B C D of 

Fig. 433. 




the body. Fig. 422, are only acted upon by the two opposite forces 
P and — P, which balance each other, while the sides A B and 
Fig 433 ^ ^ remain free from the ac- 

tion of extraneous forces ; for 
the neighboring elements of 
the body are subjected to the 
same axial strain as the sup- 
posed element A B C D itself. 
But the case is different when 
the body is bent ; for on one 
side ^ ^ of the element 
A B C D Si strain is pro- 
duced which is opposite in di- 
rection to that upon tlie other 
side C D of the element, and 
in consequence of the cohesion 




§257.] ACTION OF THE SHEARING ELASTICITY, ETC. 51J 

in A B aud CD, tlie element A B C D is subjected to the action 
of a couple. This couple is a maximum for an element which lies 
in the neutral axis ; for the element is here subjected on the side 
^ ^ to an extension, and on the side (7 Z> to a compression. 

If 8 is the strain upon a fibre at the distance e from the neu- 
tral axis, when the cross-section = 1, the strains upon the portions 
Fx, F.i, i^a . . . of the entire cross-section, which are situated at the 
distances z^, Zc^, z^ . . . from the neutral axis, are 

^Lh S,^^ S,^^ S,QtG,, 

e e e 

and the total strain in the cross-section Fi + i^a + i^s • . . is 

. Q = —{F,z, + F,z, + ...)= —^(Fz). 

B B 

Now if i^i + i^ -i- . . . is the part of the cross-section on one 

side of the neutral axis, Q is the total strain on that side of the 

neutral axis. The strain on the other side is, according to the 

theory of the centre of gravity (compare § 215), equal in intensity 

to it, but opposite in direction. 

P xe S P X 
Besides we have, according to § 235, S = ~w~> ^^ — — ~^> 

P X 

whence also Q — -^ {F^ z^ + F<i z^^ -\- , . .). 

In a cross-section, which is at a distance A B = Xi from the first 
one, the strain is 

and therefore the total force with which the piece ABE tends to 
slide upon ^ ^ is 

Q- Q.^^{F,z, + F,z,^ ., .). 

NotV" if b^ is the width of the cross-section at the neutral axis, 
the shearing force along the unit of surface in this axis is 

^o- \x, ~ b,w^^''''^ ■^''''^ '"^ ~ "1;^' 

If, therefore, the girder is not to be ruptured by a sliding along 
the neutral axis, we must put X^ — the modulus of ultimate 
strength, and in order that it shall be as secure against rupture 
by shearing as against breaking across, it is necessary that JTo shall 
be at most equal to the modulus of proof strength T, i.e. that 



512 GENERAL PRINCIPLES OP MECHANICS. [§257. 

*o = ^^ 2 {Fz). 

S {F z) is also = F^ s^ — Fc^ s.,, when F^ and i^s denote the 
areas of the portions of the entire cross-section F ~ F^ + i^s? lying 
on the opposite sides of the neutral axis, and Sx and Se the distances 
of the centres of gravity of the two portions from that axis. 

For a rectangular girder, whose cross-section i^^ = d h,we have 

"L (F z) ^ F^ Sx — -^ .-^ = — , W = — ^, and I?o = h whence 

P = ll)h 5^ and h, = l = | -^y. 

n d' 
For a cylindrical girder) whose cross-section is i^= —t~j we 

2 
have, since the centre of gravity is situated a-t a distance ^ — d from 

the centre, 

TT ^^ 2 d^ 

!> (F z) = Fi Si = -5- . -— d = --r, and, scccording to § 232, 

rrd* 
W = 77-r, and bo = d, whence 

54: 

In like manner for an elliptical girder, since W = — j — , 
jPi Si = — -— .-. % a= % a"^ b and b^ = 2 b, we have P = j 7ra Z> T, 

Finally, for a tubular parallelopipedical girder, whose cross- 
section is F = b h — bi hi (Fig. 354, § 228), we have 

„ bh' -brhi' ^ bh'-b,h,' , , , , 
F, s, = 8~^' ^ 12 \ = ^ - h, 

hence P - - (^ - ^0 (^ ^^ - ^. V) ^ 

nence 2- - 3 ^, _ ^^ ^^, 

The shearing force JT diminishes as the distance of the surface, 
in which it exists, from the neutral axis increases, and becomes 
finally null at the surface of the body, where the distance from the 
neutral axis is a maximum. The intensity of the shearing force 



258.] ACTION OF THE SHEARING ELASTICITY, ETC. 



513 



X at a given distance B = h^ from the neutral axis of the body 

P2 (Fz) 



M N, Fig. 424, is also given by the formula X 
Fig. 424. 

J 



, ,^, found 
h, W 

above, if instead of 2 {F z) 
we substitute the sums of the 
products Fx Zi, F^ z.^ , . , on 
one side oi A B CD, and in- 
stead of h^ the width h^ of the 
surface at the given distance 
^1. The sums of £lie products 
Fr, Zn, F^ + xZn + , for the other 
side is, however, equal to the 
sum of the products F^ z^^ 
i^2 ^2 • • • since the products 
of the elements, situated on 

the opposite sides of the neutral axis within the distance ± h, 

balance each other. 

E.G. if the cross-section of a girder is rectangular, we have for 

the point situated midway between the neutral axis and the limit- 

h 
ing surfaces, i.e., at the distance - from the neutral axis ' 

^(Fz)^F,s, = ^-^.in = ^\h}i\ 
and, therefore, the shearing force is 







h. 



12 



while at the neutral axis its value is JTl 






§ 258. The Shearing Force in the Plane of the Cross 
section. — As the tensile and compressive forces of the ends of ai 
element A B C D, Fig. 424, are in equilibrium, so also the shearing 
forces in this element, which form two couples, balance each other. 
Now if ^ is the length A B and ^ the height B C of the element, 
we have the shearing forces along A B and C D, ^ X and — ^ X, 
and the moment of the couple, formed by them, ^ X .^ = ^ ^ Xy 
and the shearing forces along B C and i> ^ are i^ .^ and — ^ Zy 
and the moment of the couple formed by the latter is = ^ Z . ^ — 
^K Z\ now if equilibrium exists, we must have ^ ^ X ■= ^ ^ Z, I.E., 
that X= Z. - • 

33 



514 GENERAL PRINCIPLES OF MECHANICS. [§258. 

r 2 (Fz) 
The formula X = — ; iF ^®' therefore, also applicable to the 

determination of the shearing force Z along the entire cross-section. 

It is, E.G., in a girder with a rectangular cross-section, for an ele- 

p 

ment in the neutral axis = | -r-y, and for one at a distance ± \li 



from the neutral axis = | j—^ etc. 



P 

The sum of the shearing forces along the entire cross-section, 
must of course be equal to the force P, or, if several forces act at 
right angles to the axis of the beam, equal to the sum 2 (P) of 
these forces. This can be proved as follows: if we divide the 
maximum distance e of the elements of the surface from the neutral 
axis into n equal parts, we can imagine the cross-section upon the 
corresponding side of the neutral axis to be composed of the strips 

b\ -, l>2 -, bi, -, etc., whose moments in reference to the neutral 

71 n n 



axis are 



^^ilP^'ilP'^ilh^-' 



and the sum of the latter is 



= i^' ]{lb, + 2b, + 3b,-^4.b, + ,..). 
\n j ' 

li 
In reference to the axis, which is at a distance - from the neu- 

n 

tral axis, the sum of these moments is 

= ( ^' )\2 ^», + 3 i, + 4 ^.4 + . . . ), 
\n 1 

in reference to the axis at the distance 2 -, it is 

n 

and therefore the sum of all these sums to the distance e is 

= (^)\^i + (2 + 2)^2 + (3 + 3 + 3)^3 + ...] 

= (-^)\l' . ^1 + 2^ ^2 + 3\ Z»3 + . . , + ^^^ JJ. 

It follows that the sum of all the shearing forces along cross- 
section on one side of the neutral axis is 



§259. ACTION OF THE SHEARING ELASTICITY, ETC. 



515 



B. = X.b.{l) + X,i,(l)^X,i,{l) 



+ ... 



Ph 



Wn 
P ill 



times the sum last found 



But tlie measure of the moment of flexure for this half of the 

cross-section is 

h 



'^- = ^(^^') = '[*'©' -*=(¥/-*»( 



7 i'3 A\' ^ 



©'(- 



b, + 2\b, + 3' .h -^ ^'- + n\ 1),), 



whence it follows, that the required shearing force along this sur- 
face is p _ ^^ ^'^1 

In like manner we find for the half of the cross-section, situated 

PW, 

W ' 
and finally it follows that the shearing strain for the entire cross- 

p(in+ w;) 



on the other side of the neutral axis, the shearing force R.^ = 



section is B 



W 



P, since the measure W of the mo- 



FiG. 435. 



ment of flexure of the entire cross-section is equal to the sum 
Wi + IFo of measures of the moments of flexure of the two por- 
tions of it. 

§ 259. Maxiinmn and Minimum Strain. — If the strains 
in any section are known, the strain in any gi^^en cross-section 
can be found hy employing the ordinary methods for the com- 
position and decomposition of forces. In order to find the 

strains in an element A C, Fig. 425, of 
the surface, whose plane forms the varia- 
ble angle B A O — ^jj with the longitu- 
dinal axis of the body, we decompose the 
tensions in the projections A B and B C 
of this element of the surface into two 
components, one of which acts in the 
plane oiAC and the other at right- angles 
to it, and we then combine the compo- 
nents in A C, so as to form a single 
shearing force, and the components, acting 
in a direction at right-angles to A C, so as to form a single tensile 
or compressive force. If the width of the elements A B, B and 
A C of the surfaces is unity, we can put the shearing force along 




516 GENERAL PRINCIPLES OF MECHANICSo [§ 359. 



A B, = A B . 2r and decompose it into its components A B . X 
COS. -(j) QXidi A B J. Xsim 'yp, and in like manner we can put the 
shearing force along B C, — B C , Z — B C . X and decompose 
it into its components 



— B G , X sin. i/* and B C . X cos, xj). 



8z 



The components of the tensile force B C . Q = B G . — , whose 



direction is perpendicular to -S G, on the contrary, are B G . Q cos. ijy 
and B G . Q sin. ip, and it follows that the entire shearing strain 
along A G referred to the unit of surface is 

[7=: {TB . X cos.ij) -B~C . X sin.'ip + Blf . Q cos. ip) : A Gy 
and that the tensile strain at right-angles to ^ C' is for the unit 
of surface 

V = {AB^. Xsin. ip + B~G. X cos. ij) + B~G . Q cos. a/-) : A G, 

A Ti J^ G 

But -r-y^r = COS. lb and -7-77 = sin. ib, whence it follows also that 

AG AG 

JI = X (cos. ipy — X (sin. ipy + Q sin. ^p cos. \p and 

Z7 = 2 Jf sin. -0 COS. i^ + Q (sin. ipy, or, since 

{cos. i^y — (sin. Ipy = COS. 2 ip and 2 sin. ip cos. ip = sin. 2 ip, 

Sz 
U — X COS. 2 'ip + ^ Q sin. 2 ip = X cos. 2 1/^ + ^— si7i. 2 ip and 

S Z 

V = Xsin. 2yj + Q (sin. ^Y — X sin. 2 ip -{- —- (1 — cos. 2 \p). 

The strains in the surfaces A D and G D, which together with 
the surfaces A B and G D fully limit the element A B G D, give, 
of course, equal and opposite shearing and tensile forces. On the 
contrary, for a similar element of the body upon the compressed 

side Q is negative, and therefore 

Sz 
U = X COS. 2 Ip — ^ Q sin. 2 ip = X cos. 2 ip — ^— sin. 2 ip and 

V =Xsin. 2ip-iQ(l- cos. 2 ip) =Xsin. 2ip -^(1- cos. 2 ip). 

In order now to find the values of the angle of inclination ip, 
for which the shearing force U and the normal one V assume their 
maximum or minimum values, we substitute for ip, 2 ip + fi, in de- 
noting a very small increment, and require that by it the corres- 
ponding values of U and V shall not be changed. For U =^ 
X COS. 2 Ip -\- ^2 Q ^^^- ^ V^? ^^ obtain thus a second value 
Ui = Xcos. (2 Ip -h 1^) -h i Q sin. (2 ip ^ fj>) 

= X (cos. 2 Ip COS. II — sin. 2 ip sin. fj) ■}- h Q (sin. 2 ip cos, fJt' 
4- COS. 2 Ip sin. jjl), or, since we can put cos. /-^ ~ 1, 



§259.] ACTION OF THE SHEARING ELASTICITY, ETC. 517 

Ui — Xcos. 2 ip + ^ Q sin. 2 i/^ — {X sin. 2'ip — h Q cos. 2 ip) sin. ft, 

Now if we put Ui = U, we must have X sin. 2ip — ^ Q cos. 2ip = 

and therefore ^c^ , Q « / 

sm. 2 -0 = ^r-^ COS. 2 ipf I.E., 

tang. 2 i> = ^^ = ^~^^. . 
From this it follows also that 

sin. 2 lb = —— — = ^ and 

VQ' + 4:X' V{Szy + {2X ey 

2Xe 2Xe 

and that, finally, the required maximum value of the shearing force 
Uis 

2 X' -h ^ Q' , /( RzV 

^" = VTTfx^ = ^(i '?)"+^' - ^( 2"«-) + ^ • 

In the neutral axis § is = 0, and therefore JJ,,, = X and tang. 
2 V = 0, I.E. 2 V^ = and 180°, or V^ = and 90°. For the most 
remote fibres, on the contrary, J^ is = and z — c, therefore 

Um = ^ — -TT and tang. 2 ip — oo, or 2 ip = 90° and ip = 45. 

In passing from the neutral axis to the outmost fibre, the 
angles of inclination for the maximum strain change gradually 
from and 90 degrees to 45 degrees, and the maximum strain 

varies from X^ to — . 

In order to be certain that this strain shall not become greater 
than the axial strain S, which is calculated by the aid of the for- 

mula S = TT^ and is equal to the modulus of proof strength T, 

we must make X^ at most = S, or rather 

P 1 (Fz) Pxe ^{Fz) 

If, then, in the formula V = X sin. 2 V^ + -^ (1 — cos. 2 'ip) 
we put V' + /* instead of ip and again make cos. jit = 1, we obtain 
Vi = X (sin. 2 V> COS. (jl + cos. 2 -0 sin. /^) + -^ (1 — cos. 2 ip cos. fi 

+ sin. 2 ip sin. ij) = X sin. 2 i/^ + -^ (1 — cos. 2 ip) 



-\- [X COS. 2 ip + -^ sin. 2 ipj sin. fi, 



518 GENERAL PRINCIPLES OF MECKaNICS. [^259. 

and in order that ip shall cause V to become a maximum or a min- 
imum, Vi must be = F or X cos. 2 ip + ~ dn. 2 V^ ~ 0, i.E. 

^ , 2X 2Xe ,, 

tang. I^p — ^^ = — -^ — , as well as 

* % X 

sm.2i.= :f ^^==tand cos. 2 rf, = ± -^^^f^. 

The corresponding minimum of F is 



and, on the contrary, its maximum is 



</©■ 



= || + 1/(~l +x^ 



We must require the maximum F» to be at most equal to th^ 
modulus of proof strength T or 



Ql ^ / / O ^\'2 



•/(ft)' 



^^ + V i;^"l + x^ < r. 



In the neutral axis § is = 0, and therefore tang. 2 ip = — co 
or 2 -0 = 270° and ^ — 135 or 45 degrees, and F„ = — X^, on 
the contrary, V„, — + X^. In the most distant fibre, on the con- 
trary, X is = and Q — S, and therefore tang. 2 1/^ = or 2 -0 
= or 180° and \p = or 90°, and F„ — 0, on the contrary, 
Y^^ = S. In ordinary girders the maximum strain V,,^ increases 

■ -. _ ^ Pl(Fz) , ^ Pxe _ ^. 

gradually irom X„ = — to o = ^ as we pass from the 

neutral axis to the outmost fibre. 

b W 
For a parallelopipedical girder we have 2 {F z) ^=^ -—-, W — 

8 

^T)^— l and ^ = ^5 ^^<i therefore the limit values are X^ = | . 



\2' ' 2' 



^(-i") 



P 6 P a; .^ , 
^-r and /S' = , ,o ; but in general we have X = ^—^ 

6 Pr/hV ,1 . /S':^ 12P X z 



h h' 



!"M\ 2I A^^ 12P X z , , , „ 
I - 1 — z \ and — = — T-Ti — J and therefore 



^260.] ACTION OF THE SHEARING ELASTICITY, ETC. 



519 



r„ = 



Q P X z 
6P 



+ 



A^) 






= fj^[^^W(^^y 



+ 



( ( 9 ) ~~ ^ / K^^ example, torz=l h, 



F.„ = 



v. 



dP 

2 h h 
9P_ 



~,[x-{Vx' + ilY ¥], and for x = 0, 



etc. 



Fig. 436. 



If such a girder A B, Fig. 426, is fixed at one end B^ the di- 
rections of the maximum and minimum normal forces V,„ and V 

can be represented by two systems 
of lines, which cut the neutral axis 
at an angle of 45°, and the outer 
fibre and each other at an angle of 
90°. The curyes, which are concave 
downwards, correspond to the tensile 
forces, and those which are concave 
upwards to the compressive forces. 
The steeper end of any curve cor- 
responds to the minimum and the flatter end, on the contrary, to 
the maximum forces. At the ends D and D^ both these strains 
become equal to zero, while for the ends C and Ci their values are 
the greatest. 

§ 260. Influence of the Strength of Shearing upon the 
Proof Load of a G-irder. — The capability of a girder to support 

P X 6 

a certain load requires not only that the strain S = „^ in the 




outermost fibre, but also that the shearing force X^ = 



W 

P S (F z) 

~ I. W 



m 



the neutral axis shall not exceed the modulus of proof strength T. 
In the last chapter we have repeatedly given the moments which, 
in ordinary cases, we must substitute for P x'vn the expression for 
iS' ; we have, therefore, only to give the values, which we must sub- 
stitute for the force P in the expression for X^. 

If the girder is fixed at one end and acted on by a force P 
at the other end, P can be directly employed in the formula 

X^ = — r~TF — • ^^ the beam supports, in addition, a uniformly 

distributed load, whose intensity upon the unit of length is q^ we 
must substitute for P in this expression P + q x and P -\- q l^ 



520 GENERAL PRINCIPLES OF MECHANICS. [§360. 

when we wish to determine the maximum value of X^. If, on the 
contrary, the girder is supported at both ends and sustains at the 
distances Ix and l<^ = I — l^ from the points of support a load P,, 

we must substitute for one portion of the beam j P, and for the 



other V P instead of P in the formula for Xo, in order to find the 



I 

shearing force in the neutral axis. If, on the contrary, this girder 

sustains an equally distributed load q I, each of the points of sup- 

a I 
port bears %-, and the shearing force of the whole cross-section at 

any point at the distance x from the points of support is P = g' 

/- — xy The latter is = in the middle, where x = -, becomes 

greater and greater towards the end, and at the point of support 

If a girder, supported at both ends, sustains a load, which is 
equally distributed over a part c of its total length, while the other 
portion ^ — c is not loaded, the point of support of the first por- 
tion bears a part q c (l — ^1 of the total load q c and that of the 

c^ 
second portion a load ^y, and the vertical shearing force at the 

distance x from the first point of support is 

The value of the latter becomes for a; = c, — ^-y, and this value 
remains the same for any distances a? > c. If the load covers 
exactly one-half of the girder, i.e. if c = ^, we have 

-^ = 9' (-g- - icj or for ic = -, P = - ^. 

Fig. 437. If, finally, the girder A By 

Fig. 427, bears a load p I equal- 
ly distributed over its entire 
length I and a load q c equally 
distributed over the length A C 
= c, the reactions of the points 
of support are 

B.= ^ + q[c- 2-,) and R. = -^- + ^, 




§360.] ACTION OF THE SHEARING ELASTICITY, ETC. 521 

whence it follows, that the vertical shearing force at the distance 
A = X from the point of support A is 

ioT z — c the latter expression becomes^ (9 "~ ^) "~ 97^ ^^^ ^^^ 
any distances re > c it is 

^ ~ c\ — ^— - in 6' is = 
for c'* + — ^ c = £. x^^ I.E., for 

If, in general, at a point of the girder the shearing force is 
P = R — q x^ WQ have for the moment of flexure 



M= Rx - 



qx 



=¥(^-4 



2 2 \ ^ 

This, however, for x = x, i.e., for x = — , is a maxi- 

q q 

mum, in which case P becomes = ; the moment of flexure of a 
girder becomes a maximum for the same point at which the verti- 
cal shearing force is = 0, and in the foregoing case c gives that 
length of the load q c, for which the moment 



becomes a maximum, and it is then 



2 
(i? 4- q) & 



2 

These formulas are applicable to girders for bridges, where q c 
denotes the intensity of the moving load. 

The shearing force X^ — — - must be specially consid- 

ered in the case of bodies of uniform strength, the cross-section of 
which, according to what we have seen above (§ 253), might in 
some parts be infinitely small. For example, for the parabolic 

i P 
girder in Fig. 406, we have X^ — T ^=^ ^ . r-^> and therefore, the 

p 

necessary cross-section at each end is F^ = h^h^^ = ^ 7=^, in which 

T denotes the mod ulus of proof strength for shearing. 



522 GENERAL PRINCIPLES OF MECHANICS. [§261. 

§ 261. Influence of the Elasticity of Shearing upon the 
Form of the Elastic Curve. — We have yet to determine what 
influence the elasticity of shearing has upon the form of the elastic 
curve or upon the form of the neutral axis of a loaded girder A B, 
Fig. 428. According to the formula P — t F C\ in which C de- 
notes the modulus of the elasticity of 
Fig. 428. shearing and F the cross-section of the 

beam, the inclination the beam A^ B pro- 

duced by the shearing force is i = ~, 

and, therefore, the corresponding deflec- 
tion of the end A^ of the girder, whose 
length A^ B ~ l,\^ 

To this must be added the deflection A^ A = a^, produced by 
the flexure of the beam, and which, according to § 217, is «2 — 

F r 

^ -^ ', the total deflection of the girder is therefore 

BC^=AQA = a = ai + a2 = -jy ( , ^ + w-jp )• 
For a parallelopipedical girder §„ = J, 2 [F z) = -^- and TF= 
y; consequently 




8 



_4 PTr ^\F/hy-\ 



or, assuming -^ = 3, 



4: FT 

a — 



hh' F 



v*tm 



4: p r 

E.G., for ^ := 10 7i, we have a = 1,01125 . , ,„ „ , if then the 
' d¥ F' 

girder is ten times as long as thick, the deflection at the loaded 

end, due to the shearing force, is so small compared with that due 

to the flexure of the girder, that in most cases we can neglect it. 

In order to determine the modulus of elasticity of a girder A B, 

we load it first with a small weight F at the greatest distance I, and 

afterwards with a large weight Fi at a smaller distance /i from the 

point of support B, and we observe the corresponding deflections 

u and tti of the length I of the girder. Now we have 



'• 262.] ACTION OF THE SHEARING ELASTICITY, ETC. 



523 



a = 



Pl^{Fz) PI 



K W C 



+ 



3 WE 



and 



p, i^{F z) ^ Pa. 



+ 



P. I.' {I - I) 



b, W G \ 3 WE ' 2 WE 
In order to eliminate C, divide the first equation by P and the 
second by Pi and subtract the equations obtained from one 
another. Thus we obtain 

P P,~ WE\ 3 2 J ~ W E\3 2 "^ 6/' 

and therefore the modulus of elasticity for tensile and compressive 

forces IS E — - — ^ ^VT^^ ( o 

•^ (a P^ — a-i P) n \3 

With 



2 



(aP,- a, P) PF\3 2 ' 6, 
the aid of this expression and the formula for a, we 
determine the modulus of elasticity for shearing by the formula 

PI S^ (Fz) E 



C = 



^ 



3 W E a - PI' 



' § 262. Elasticity of Torsion. — ^^In order to investigate the 
theory of the hoisting or torsion of a body (see § 202), we can again 
begin with the case of a body H C D L, Fig. 429, fixed at one end, 

but, in order to avoid any 
Pig. 429. complex change of form, we 

must assume that the free 
end is acted upon by a couple 
(P, — P) whose plane AHB 
coincides with the plane of 
rotation of the axis C D. 
Let us imagine the body to 
be composed of long fibres, 
such as H K, which, in 
consequence of the torsion, 
assume the form of a helix, 
by which H K comes into 
the position L K and the 
Whole base is turned through an angle ^(7 Z — a. If the portions 
Hx Ki, Hc^ K<i, etc., of the fibres, whose lengths are unity and whose 
cross-sections are F^, F^ etc., undergo a lateral displacement through 
the distance H^ L^ = er,, R^ Z^ — o^ etc., we can put, when the modu- 
lus of elasticity for shearing is C, the corresponding shearing forces 
Si = (Ti Fi C, Si = (72 F^ C, etc. Now if the corresponding angle 




524 GENERAL PRINCIPLES OF MECHANICS. i;§ 263. 

of torsion is ^i L^ = H^ Lc, = ^ and if the distances of these 
fibres from the axis G D of the body are H^ = z^^ H^ = z^, we 
have Gi = fl> Zi, a^ = (p Z2 . . . ; hence the strains are Si — cj) C Fi Zi, 
JSa = (}) C F2 Z2 . . ., and their moments are 



S,zr== (pCF, z,\ S,z, = (l>CF, z, 



2 
2 • • • 



All the forces 8x, /S^s • . • of a cross-section H^ L^ must in any 
case balance the couple (P, — P)\ if then a is the lever arm A B 
of this couple 01 P a its moment, we can put 

P a= S,z, + S,z, + .., = ^ C F,z,^ -\- (p C F,z,^ + ... 

Now if we designate the geometrical measure Fi Zi^+ F^z^ + .... 
of the moment of torsion by W, we have P a = (p O W. 

But the angle of torsion for the entire length D = I of the 
body is a = (p I, therefore we can put 

1) Pa = ^L^-K^ ovPal=aCW, 

6 

and the angle of torsion 

_. _Pal 

As we have done previously (§ 215), we can call W G the 
moment of torsion, and consequently TFthe measure of the moment 
of torsion, and we can then assert, that the moment of the force P a 
increases directly as the angle of torsion and inversely as the length 
of the body. 

The work done in producing a torsion equal to the angle a is 

L -~.aa- ^^ - ^ ^^; 

for the space described by the force P, which causes it, is a a. 
These formulas hold good for prismatical bodies alone, for bodies 
with other forms we must substitute in them instead of the ratio 

-T^ a mean value of it. 



" § 263. Moment of Torsion or Twisting Moment. — The 

measure W = F^ z^ + F^ z^ + ... of the moment of torsion can 
easily be calculated, according to the rule explained in § 225, from 



§363.] ACTION OF THE SHEARING ELASTICITY, ETC. 



525 



\ 



y 




the measure of tlie moment of flexure for the same cross-section. 

If, for example, Wi is the measure of the moment of flexure of a 

surface A B D, Fig. 430, re- 
ferred to an axis X X and W^ 
the same in reference to an axis 

Y Ysit right angles to the first, 
we have for the measure of the 
moment of torsion in reference 
to the intersection of the two axes 
W= Fi + TTo. 
For a shaft with a square cross- 
section A B D E, Fig. 431, we 
have, when h denotes the length 
of the side A B=D E, according 
to § 226, the measure of the mo- 
ment of flexure in reference to each axis X X and Y Y 

l¥ _ V_ 
12 ~ 12' 
and consequently the measure of the moment of torsion is 

and the moment of the force 

Pa = —^=-^=0,1667-^ 

For a shaft with a rectangular cross-section {b h) we would 
have, on the contrary, 

P « = ^Mf_ti!) a = 0,0833 ^AAi^;^*!^. 

Fig. 431. P Fm. 433. ^_q 

H 



-XI 



Wr= W,= 





P Q 

For a cylindrical shaft with circular cross-section A B, Fig. 
432, whose radius is C ^ = r, the measure of the moment of 

flexure in reference to an axis X X or 1^ J^ is (according to § 231) 



§26 GENERAL PRINCIPLES OF MECHANICS. [§ 36a 

and therefore the measure of the moment of torsion in reference to 
the point C in that axis is ■ 

77 T* 
W=2 Wr= —-, 

Now if the twisting couple (P, — P) acts with an arm H K 

Ob 

— a, or each of its components with an arm C H — C K — ^y 
we have 

If the shaft is hollow and its radii are r^ and ^s, we haye the fol- 
lowing formula : 

a 77 (ri* — r/) G {r,' — r,') G 

F a = ^ — ^f~ — - — = 1,5708 a ~ — - — 

Z I I 

The torsion of a shaft ABM, Fig. 432, is generally produced 

by two couples (P, — P), (ft — §), which balance each other, 

and therefore, instead of /, we must substitute not the entire length 

of the shaft, but the distance between the planes in which the two 

couples act; it makes no difference, however, whether we make the 

moment of torsion equal to the moment of the couple (P, — P) or 

to that of the couple (§, — Q). If we denote the arm H K oi the 

couple (P, — P) by a, and the arm N oi the other couple 

( Qi — Q) by d, we have 

P a = Qo — — -. — . 

The foregoing theory gives us for bodies limited by plane sur- 
face moments of torsion, which vary somewhat from the exact 
truth ; for we suppose, in calculating them, that the bases of the 
prism subjected to the torsion remain plane surfaces, while, in re- 
ality, they become warped. According to the researches of Saint 
Venant, Werthheim, etc. (see the " Oomptes rendus des seances de 
Tacademie des sciences a Paris," T. 24 and T. 27, as well as " ITn- 
genieur," Nos. 1 and 2, 1858; in German in the " Civilingenieur,'^ 
4 Vol., 1858), we have for a square shaft 

Pa — 0,841 — —. — = 0,1402 — = — , 

D / t 

in which h denotes the length of the side of the square cross-section^ 
For bodies, the dimensions of whose cross-sections differ very 



g263.] ACTION OF THE SHEAEING ELASTICITY, ETC. 527 

much from each other, these variations are greater ; e.g., for a pris- 
matical body with a rectangular cross-section, whose width is h and 
whose height is h, we have 

W = W, + TF2 = — + — = 5l_^ \ and therefore 

_ aWG _ ahh {V + h') G 

Now if this formula requires a correction, when ^ = J, in which 
case P a =^ ^ , it is natural to expect that when h differs ma- 
terially from A, in which case the surface of the sides will become 
more warped, it will no longer be suflBciently accurate. In fact, 
taking into consideration the warping of the surfaces, we find by 
means of the calculus 

aWV G 
Fa 



3 {v + le) V 

and according to the later experiments of Werthheim, the mean 
value of the required coefficient of correction is = 0,903 ; conse- 
quently we must put 

Pa — 0,903 ^-772 7n~7 — 0,301 



3 {h' + If) I ' {h' + ¥) I' 
If h is very small compared to Ti, we have 

a li h' G 



P a=^ 0,301 



I 



If the angle of torsion is given in degrees, putting a = 



a^TT 



180' 
= 0,017453 a% we obtain 

1) for prismatic girders or shafts with a circular cross-section, 
the diameter of which is d = 2 r 

rat- —^-L-~^ ^^-180^2^-180^32^ 

= 1,571 ar' G = 0,0982 a d' G = 0,02742 a' r' G 
= 0,001714 a' d' (7; 

2) for prismatic girders, axles or shafts with a square cross-section^ 
the length of whose side is I, when we neglect the coefficient of 
correction, 

Pal= ^? = 0,1667 o J' C= °° ^^^^^ = 0,00391 a' V C. 



528 GENERAL PRINCIPLES OF MECHANICS. [§264. 

Inversely we have 

a = 0,637 ^^^ = 10,18 ^^ = 6 -^, and 

„ Pal Pal Pal 

«° = 36,4 ^;r^= 583^^ = 344 ^j^. 

The vakies for C must be taken from Table III. in § 213. 
Hence we have, e.g., 

1) For cast iron, C = 2840000, whence 

Pal = 77900 a" r* = 4867 a° d' = 8264 a° b* and 

a« = 0,00001281" ^4^ = 0,0002053° "^^^ 



nz 0,0001211 



r' ' d' 

Pal 



b' 
^) For lurought iron, G = 9000000, whence 

P al = 246780 a" r* = 15426 a** 6?' = 26190 a" J* and 

a° = 0,00000404° ^^^=0,0000648° ^^^=0,0000382° 4r^. 

3) For woofZ, C = 590000, 
Pal = 161800 a° r' == 1011 a° d' = 1712 a° ^* and 

P aJ Pal P n 1 

a' = 0,0000617" ^-^ = 0,000988" -~ := 0,000583" -^. 

Example — 1) What moment of torsion can a square wrought-iron shaft 
10 feet long and 5 inches thick withstand, without suffering the angle of 
torsion to become more than ^ of a degree ? Here, according to this table, 
we have 

635 
Pa = 26190 . i . T?r— i o = ^4102 mch-pounds = 2842 foot-pounds. 
10 . 1-* 

2) "What is the amount of torsion sustained by a hollow cast-iron shaft, 

whose length is ? = 100 inches and whose radii are r^ = 6 inches and 

7*g = 4 inches, when the moment of the force is P « = 10000 foot-pounds ? 

Here a° (r^* — rJ) 

Pa = 77900 —^--i —, 

consequently 

Pal 10000.12.100 



77900 (r/ — 7-3*) ~ 77900(62 4. 42) (63 —42) 

_ 120000 

~ 779 . 52 . 20 

1500 , 
^ ^ = degrees = 8,887 minutes = 8 mmutes 53 seconds, 

§ 264. Resistance to Rupture by Torsion. — If in a prism 
C K L, Fig. 433, twisted by a couple (P, — P) the shearing force 
per unit of surface at a certain distance e from the axis C /> is = ^S', 



264.] ACTION OF THE SHEARING ELASTICITY, ETC. 



529 



the shearing force at any other distance Zi is = - S, and its mo- 



Fig. 433. 




S 



s 



ment is = — S, and for a 
e 

cross-section Fi it is 

e e 

in like manner the moments 
of the shearing forces of other 
cross-sections F^, F^ . . ., 
which are at the distances 
z^y 2^3 .. . from the axis C D, 

are — F^ z^^ — F^, z^, etc.; 
e e 

hence the total moment of tor- 

sion of the body is 

S 



Pa = — F, z^ + — i^2 ^2' 4- — -^3 ^3' + . . . 
e e e 

= — {F, z,' + F,z,' + ., .). i-E. 

1) P a = , or P a e = S W, and — = — ^. 

Substituting for S the modulus of proof strength T for shearing, 
and for e the greatest distance of the elements of the cross-section 
from the neutral axis, we obtain in the formula 

2) P ae = T W 2in. equation for determining the dimensions 
of the cross-section, which the body must have if it is not to be 
strained at any point beyond the limit of elasticity. If, instead of 
the modulus of proof strength T, we substitute the modulus of 
rupture K for shearing, we obtain the moment Pi a, which will 
break the body by wrenching ; it is 

3) Pia = 

e \ 

For a massive cylindrical shaft, whose diameter d = 2 r,we 
have 

W 

e 



-^— = -X—, and therefore 
2r 2 



Pa = 

Pia = 
34 



TTr'T 7T d' T 



2 
2 



16 
rrd' X 

16 



= 0,1963 d' T, and also 
= 0,1963 ^ K. 



530 GENERAL PRINCIPLES OF MECHANICS. [§264. 

If the shaft is liollow and the diameters are dx = .2 r^ and d<i = 
2 Tif in which case 

W _ 7T(rx' -r;-) 



e 2 Ti 



-, we have, on the contrary, 



in which F = — ^-^j — denotes the cross-section of the body. 

Por a prismatical body with a square cross-section, the length 
of whose side is Z>, we have 

W = -^ and e = -^hV2 = I?V^, whence 

— =: ^-= r= — -= and P « = ^^ = 0,2357 I' T, 

e 6V^ 3|/2 3|/2 

If in the fundamental formula Pa = (j) C W of § 262 we substi- 

(7 tayia, o 
tute (b — - = — ^ — , in which e denotes the distance of the most 

^ e e 

remote fibre from the axis of rotation CD and (5 the angle H K L, 
which this fibre has been turned from its original position by the 
torsion, we obtain 

P a e — C W tang. 6 ; but we have also 

P a e = 8 W, hence 

S = C tang, d, and therefore 

T 

T = C tang. 6, or tang. 6 — —, 

in which d denotes the angle of displacement, when the strain 
has reached the limit of elasticity. 

The mechanical effect, which is required to twist the shaft 

through an angle a, is, according to § 262, L = o w n ^ ^^^ there- 

S W S^ W I 

fore if we substitute P a =^ , we can put L — -^ -?r~2~ ^ iii 

e C 2 e 

which 8 denotes the maximum strain. 

At the limit of elasticity 8 — T; hence it follows that the me- 
chanical effect necessary to twist the body to the limit of its elas- 
ticity is 

^~ C ' 2e'' 



§264.] . ACTION OF THE SHEARING ELASTICITY, ETC. 531 



77 r 



For a prismatic body with a circular cross-section W 

and e = r, whence 

J _ T" 7^ - I! V 

2(7' 2 ~4(7' 

and, on the contrary, when the cross-section is a square 

^* ¥ 

W = ^r and e^ = -^, and therefore 

o 2 

2 (7 • 3^'' 6 6 

Now ?r->^ = - ^ ^ = -^- is the modulus of resilience for the 

2 (7 2 C 2 -^ 

limit of elasticity ; hence we have for the cylinder L = I A V, and 
for the parallelopipedon L = I A V. 

The tvorJc done in both cases is proportional to the volume of 
the body alone (compar^ § 206 and § 235). 

We can also put for the mechanical ejffect necessary to rupture 
of the body by wrenching L = ^ B V and I ^ F, in which B 
denotes the moduhis of fragility for wrenching. 

If we assume with General Morin for all substances 

^ =: tang, 6 = 0,000667 

or the angle of displacement 6 — % min. 18 sec, we obtain for 
cast iron 

T = 200000 . 0,000667 = 134 kilo. = 1906 lbs., 
therefore, when we employ the French measures 

Pa — 26,3 d^ = 31,6 V" kilogr. centimeters, 
and, on the contrary, when we employ the English measures 

Pa — 374 fP = 449 W inch -pounds. 
Under the same conditions we have for wrought iron 

T = 630000 . 0,000667 = 420 kilo. = 5974 lbs., 
and therefore 

P a = 82,4 d^ — 99,2 b^ kilogram centimeters, 
or 

P a = 1173 d' = 1408 b' inch-pounds. 
Likewise under the same conditions we have as a mean for 
wood 

F = 41650 . 0,000667 = 27,8 kilogr. = 395 lbs., 
whence 

Pa — 5,46 fZ' = 6,55 F kilogr. centimeters, 
or 

P a = 77,5 d' == 93,1 ¥ inch-pounds. 



532 GENERAL PRINCIPLES OF MECHANICS. • [§ 265. 

The coefficients of these formulas are correct only for bodies 
at rest or for shafts, which turn slowly and smoothly ; for common 
shafts we give double security, i.e., we make the coefficients but 
half as great. When their motion is very quick and accompanied 
by concussions, we are obliged to make the coefficient but one- 
eighth of those given above. 

Example — 1) The cast iron shaft of a turbine wheel exerts at the cir- 
cumference of the cog-wheel upon it, which is 6 inches in diameter, a 
pressure of 4000 pounds. Required the thickness of the shaft. Here the 
moment of the force is P a = 4000 . 6 = 24000 inch-pounds, and conse- 

374 
quently'the diameter of the wheel, when we put P a = -^ ^^, is 

d = y ~r^~ ~ ^'^^ inches. 

If the distance from the cog-wheel to the water-wheel is Z = 48 inches^ 
we have, according to the foregoing paragraph, the angle of torsion 

24000 . 48 
= 0,0002053° l^,: = 0,367° = 22'. 

' 5,04^ ' 

2) A force P = 600 lbs. acts with a lever arm a = 15 feet = 180 inchea 
upon a square fir shaft, while the load Q acts with an arm of 2 feet at a 
distance I = Q feet = 72 inches in the direction of the axis ; how thick 
should the shaft be made and what is the angle of torsion ? 

In order to have quadruple safety, we must put 

93,1 5' 
Pa = QOO . 180 = 108000 = —^, — , 

4 ' 

hence the width of the side is 



;/4. 108000 ,^^^ . , 
d = y — — = 16,68 inches, 

and the angle of torsion is 

108000 72 
a" = 0,000583 - ' = 0,0586 degrees = 3^ minutes. 



(16,68)' 







CHAPTER IV. . 

OF THE PROOF STRENGTH OF LONG COLUMNS OR THE RESIST- 
ANCE TO CRUSHING BY BENDING OR BREAKING ACROSS. 

§ 265. Proof Strength of a Long Pillar Fixed at One 
End. — If a prismatic body A B (I), Fig. 434, is fastened at one end 



§ 265.] 



PROOF STRENGTH OF LONG COLUMNS, ETC. 



533 



B and acted upon at the other by a force P, whose direction is that 
of the longitudinal axis of the pillar, the relations of the flexure. 

Fig. 434. 




under these circumstances, are very different from what they are 
where the force acts, as we have seen in § 214, etc., at right angles 
to this axis. The neutral axis A B (II) assumes in this case 
another form ; for the lever arm of the force P is represented by 
the ordinate M = y and not by the abscissa A M = x, and its 
moment is not P x, but P y ; consequently the radius of curva- 
ture K = r \^ determined by the expression 

_ WE 

while, according to § 215, for a bending force acting at right 

angles to the axis we must put 

_ WE 

^ ~ Px 

At the point B, where the pillar is fastened, y becomes the de- 
flection B G 



WE 
a, the radius of curvature r = -^ — is a minimum 

Pa 

and the curvature itself a maximum. On the contrary, at the point 

of application A, where y = 0, the radius of curvature is infinite 

and the curvature itself null. 

If we denote by 6? the arc, which measures the angle K Oiof 

curvature of the element Oi = cr of the curve, we have r 



a 

d' 

and therefore P y o = W E 6; and if (i° is the angle of inclina- 
tion Oi JV of the same to the axis A C, we can put the element 
iV of the ordinate — v = a (3, and therefore 

P y V = W E (3 6, and in like manner 

Pll{yv) = WEI. {(3d). 



534 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 365. 



In order to find the sum 2 (y v) for the arc A 0, let us substi- 
tute ioT y,v,'Z v,^ V , . . n V in. the above equation. Thus we 
obtain I>{yv) = v'^{y) — v{v + %v-\-Zv + ,.. + % v) = v 

or since n v — M — y, 



2 



% 



f 



^{yv)^ fsindPl{yv) = iPy\ 

In like manner, to find 2 (/3 6), we substitute for (3 successively 
P, 13 -\- d, (3 + 2 6 ... [3 + n 6, and complete the summation as 
follows : 

^{f3 6) = 6-E{(3) = 6{f3-^f3 + 6-^f3^2d + .,.-^(3 + n6) 
== (5 [?^ /3 + (1 + 2 + 3 + . . . + ^) fl 

If the angle of inclination at^, = a, we can -put (3 + n 6 = a, 
and therefore 

l{f3 6) = {a- (3) (^[3 + "^^j = i(a- f3) {a + [3) = -J (a' - /3^), 

whence 

JVEl.{l3d) = iWU{a' - (3'), and finally 
Py-= WB{a' - 3'). 
For the end B,y = a and [3 = 0, and therefore 
Pa'' = l^^^a'and 
P{a' -f)= WEi3\ 
from this we obtain the tangential angle 

^ ^ ^ WE ' 
From (3 and the element N — v of the ordinate we obtain 
the element of the abscissa 



N0 = ^:=^ 



V 





r 



v\/ 



WE 



V 



/ 



P (a'- f) Va' 

^p V 



y 



\/^ 



E 



or 



/2* 



WE Va'- y' 
If with the hypothenuse C ^ = ^ of the right-angled triangle 

B C B, Fig. 435, whose altitude is B B = y 

and whose base is C B = Va^ — y'^, we de- 
scribe an arc A B, we have for the element 
B = ^ the proportion 



Fig. 435. 
B 




BO C B 



^ 



a 



BN 
whence 



C B'^'^' V ~ 



Va' 



y 



2> 



§265.] PROOF STRENGTH OF LONG COLUMNS, ETC. 535 

=: —and 



Va'-y f^ 



^y -:r-—- = — , as well as 



/ 



^ 2 (^) - ^ 2 {^y 



WE ^'^ a 
But 2 (^) is the sum of all the elements of the abscissa and is 
= X, and 2 (i/j) is the sum of all the elements of the arc A B and 
is equal to the arc A B itself; therefore we have also 



./ P arc ^ ^ . .y 

^ W E a a 



The abscissa of the elastic curve A B, Fig. 434, II, is therefore 

2) X = y — ^- . sin. -, 
and its ordinate is 



3) y = a sin. ( x \/ -^y-^)- 



If X = A B — A (7=Z, the length of the column, we have 
y = the deflection B C ~- a; therefore 

a .= a sin. (l y ^j^-^Jy i-e., sin. \^l y ^^^ = 1, 

whence 

/ p n 

I y -jj^-^ = -^, from which we obtain the bending force 

Since this formula does not contain the deflection a, we can 
assume that the force P, determined by it, is capable of holding the 
body in equilibrium, however much the body may be bent. This 
peculiar circumstance is owing to the fact that the increase of 
the flexure is accompanied not only by an increase of resistance, but 
also by an increase of the lever arm a, and consequently of the 
moment P a of the force. 

The force necessary to rupture the pillar by breaking it across, 
is therefore 

WE 



P = {yi) ^^'^= ^'^^'^^ 



V 

Remark. — If we substitute in the formula y = a sin. ix i/ — - ), P = 

\ ' WE/ 



WE, we obtain the following equation of the elastic curve for this 
case of the action of a force 



(fj 



536 



GENERAL PRINCIPLES OF MECHANICS. 



[§366. 



. /TT X\ 



Substituting in this 


.X = 






-f 
c 


21 


3Z 


4.1 


51 


6 I, etc., 


we obtain 


•y = 


a 





— a 





a 


0, etc. 



If, then, a column, whose length is I, is increased any amount in length, 

a force P = ( — ) WE will bend it in the shape of the serpentine line 

A B A^ B^A^.. ., Fig. 436, which is composed of a number of similar arcs 
^5 and is cut by the axis J. Xat the distances J^ J. 1,^.^.3, 
. . ., and at the distances A C, A G^, A Cg, the curve is 
at its maximum distances C B = a, G^ B^ = — a, G B^ = 



Fig. 436. 
A 



a 



from this same axis. 



§ 266. Parallelopipedical and Cylindrical 
Columns. — For a parallelopipedical column, the 
greater dimension of whose cross-section is b and the 

smaller one is h, we have }V= -^^ (see § 226), and con- 

sequently the force necessary to rupture the same 
OiH^i by breaking it across is 

The resistance of a paraUelopi2Mdon to breaking 
across is directly proportional to the width b and to the 
'' f fibe (h^) of the thickness or smaller dimension h of its 
t ross-section and inversely proportional to the square 
(r) of the length.' 
For a cylindrical pillar, whose radius is r or whose diameter is d, 



4 



'nd' 
64 



(see § 231), consequently we have 



- (-Y - 



TT 

4 ^ = 16 



r' E 



7T 



d' E 



= 0,1211 



r 256 

d' E 



1,9381 . 



r' E 



I' 



Therefore the {(reacting) strength of a cylindrical column, by 
which it resists bending or breaking across, is directly ^proportional 
to the fourth power of its diameter ayid inversely proportional to the 
square of the length. 

For a hollow column, whose radii are r and fi, and whose diam- 
eters are d and d^ — 11 d, we have 



366.] PROOF STRENGTH OF LONG COLUMNS, ETC. 



537 



16 f ~ 256 V 

If the column A B A, Fig. 437, is . not fixed at the lower end 
A, but only stands upon it, it will bend in a symmetrical curve, 
each half B A and B A^ haying the form of the axis of a column 
fixed at one end (Fig. 434). The above formula can be applied 

directly to this case by substituting-instead of Z ; Z of course denotes 

the total length of the pillar. The proof load is therefore four 
times as great as in the first case, and it is 

This case of flexure occurs when, as is represented in Fig. 437, 



Fig. 437 



Fm. 438. 





I. and III., the ends of the pillar are rounded or when they are 
movable around bolts. An example of the latter case is the con- 
necting rod of a steam engine. 

, If a pillar is fixed at both ends, as is represented by ^ ^ B^y 
Fig. 438, I. and III., its axis will be bent m a curve B A C A, B,, 
Fig. 438, II., with two points of inflection A and ^„ and in which 
the normal case of curvature is repeated four times, substituting, 

therefore, in the formula for the normal case -, instead of L we ob- 

4 



538 



GENERAL PRINCIPLES OF MECHANICS. 



t§ 366. 



tain the proof load of such a pillar fixed at both ends 

1 W HJ =i ■ 

3 I 



P = 



^i^il^jWE=^'-lE = 



7T 



/7* 

16 r ^' 



According to Hodghinson^s experiments, the proof load is only 
twelve times as great as in the normal case, while according to the 
above formula it would be sixteen times as great. 

The principal example of this case of flexure is that of the 
'piston rod of steam engines, etc. 

If, finally, a column A B, Fig. 439, is fixed at one end B and 
at the other prevented from sliding sideways, 
the proof load P is eight times as great as in 
the normal case, or 




W E = 



6 r 32 r 

The force which is necessary to crush a 
column, whose cross-section is F and whose 
modulus of rupture is K^ is given, according to 
§ 205, by the simple formula P — F K. 
If we put this force equal to the force 



P 



\2ll 



^VWE 



necessary to produce rupture by breaking across 
in the normal case, we obtain the equation 

For a cylindrical pillar, whose thickness is d, in which case 

F 16 

r=p = -^, it follows that 
W d 



2? 



\ = l^- 0,3927 /f . 



For cast iron E 



8 ' E ~ ^'^""' ' K 
17000000 and K = 104500, hence 



■i/~ = V 162,68 = 12,8 and -, = 5, 
^ E d 

For wrought iron E = 28400000 and E = 31000, hence 



/■ 



E 



I 



— = V die = 30,3 and -, = 12. 
E d 

Finally for wood we have as a mean 

E = 1664000 and E = 6770, hence 



§267.] PROOF STRENGTH OF LONG COLUMNS, ETC. 539 



V 



^ = V 24:6 .= 15,7 and ^ = 6. 



If a column is free at both ends, the values of -^ are twice as 

great as those found aboye. 

When the ratio of the length to the thickness is that just given, 
the resistance to breaking across is equal to that of crushing, and 
it is only when the pillars are longer than this, that the resistance 
to hreahing across exceeds the resistance to crushing. In this case 
the dimensions of the cross-section are to be calculated by the 
above formula. 

Example — 1) The working load of a cylindrical pine column 13 feet 
long and 11 inches thick, assuming 10 as a factor of safety, is 

P= ^jZ?k = 0,4845 (IIX . 166400 = 80620 . 0,7061 = 56900. 
64 V 10 V12/ 

2) How thick must such a column of cast iron be made, when its length 

is to be 20 feet and the load 10000 pounds ? Here, if we put instead of B^ 

— = 1700000, we have 



y ^3 1700000 r 



, , .. . ^ „ . /640000 . 240^ 

a 



7rM700000 y 31 . 1700000 



/ 240^ / 240 

According to the formula for the strength of crushing 






d = \/ ^-^, 



K 



or, substituting ^x = 10400 pounds m the calculation, we have 
, /4 . 10000 ,r~^m~" i/^O" ^ ^^^ . , 

' TT , 10400 ^ TT . 104 '^ 13 TT ' 

If the length of the pillar does not exceed 10 . 1,106 = 11,06 inches, the 
required thickness would then be but 1,106 inches. 

(§ 267.) Bodies of Uniform Resistance to Breaking 
Across. — If a pillar A B, Fig. 440, fixed at one end, is so shaped, 
that in all its cross-section the strain is the same, a solid of uni- 
form resistance is formed, which requires the minimum amount of 
material for its construction (see § 208 and § 253). The cross- 
section of such a body is certainly a maximum at the fixed end B, 
and it decreases gradually towards the end A. The law of this 
decrease is found as follows: denoting again by a; and j/ the co- 
ordinates of a point in the axis of the column, by a the tangen- 



540 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 267. 



tial angle MA for this point, by TF the measure of the moment 
of flexure, by z the radius Oi of the column at this point and by 
S the strain at the surface A Ox B^, which is there- 
fore that at the point Oi of the cross-section through 
0, we have 

^ Mz P y z , „ __^, _ 




M 



„ WE Trr r,d tauq. a 

^ r dx ^ 



(see § 218), whence 

^ T7 ^ tana, a . , dy 

iS = — E z Y^ — or, smce tana, a = -^, 

d X ^ dx 

S d y — — E z tang, a d tang. a. 

But, since for a circular cross-section — = 

z 



4 ' 



8 



p y — — ^^ or J S z^ = P y, and we have 



W 



7T Z 

dy = J -pd (z^) = —r- -p- z^ d z and Sd y = —r- -^ z^ d z, 



whence 



Z IT 8' 



z d z = — tang, a d tang. a. 



4 PE 
By integration we obtain 
_8^ 
4 "^ PE 

and, if we denote the radius of the cross-section at B by r, we have 
8' 



rr -^-^ z" = Const. — tang,^ a, 



I TT 



P E 



{r^ — z'') — tang,^ a, since a = ; hence 



tang. 



Putting tang, a = -^ 

a X 



'.a = 8i/^^,Vl^^z\ 



4.P E 



I TT 



8_ z'dz 
P ' dx 



; we obtain 



f A 



E z'dz 



4P 



d X 



= Vr" — z^ and 



dx 



^ 4.P 



z'd 



Vr' - 



z ../dnE 



4P 



u^ du 



2 

when - is denoted by u. 



§ 267.] PROOF STRENGTH OF LONG COLUMNS, ETC. 641 

But 

U^ 1 — u^ " 



+ .A 5 = - VI - u" + 



1-tt' Vl - u' Vl -u'T '^ Vl ^u' 

and therefore 

f u'du _ r /* du 

J a/ 1 ~ ~ J y'^ - u\du + J 



|/r=:i? -^ r . - . . c. . -r ^ ^^^ 



w 



= - ^uVl-u\+ iJ- 



d u 



Vl - u 



2 



= — ^ u Vl — u* + ^2 sinr^ u. 

(See the Introauction to the Calculus, Art. 27 and 26.) 
Hence we have 



I = ~ r^ Y p and that the proof load is 



For X = I, z — r, the radius of cross-section of the base, for 

which sm.~^ - = sin~'^ 1 = tt and 
r 2 

z Vr^ — z^ = 0. Therefore it follows that 

\2l/ 16 ^ ~ 4 \2l/ 4 ^' 

that is, three-fourths of the proof load of a cylindrical pillar, whose 
radius is r (compare § 265). Consequently the radius of the base 

of a column of uniform strength is = ^ | == 1,075 times the 
radius of a column of the same length whose proof strength is the 
same. 

Comparing the abscissa x with the total length I of the column, 
we obtain 



segment of a circle, whose radius = 1 and whose chord = — . 



2x 



If, then, we regard - v as the area of the segment of a circle, we 

can determine, by means of a table of segments (see the Ingenieur, 
page 152), the corresponding angle at the centre, and from it we 
can calculate for a given abscissa x the corresponding radius of the 



542 GENERAL PRINCIPLES OF MECHANICS. [§268. 

cross-section z — r sin. ^ : e.g., for x = I- I, — ,- = - — 0,3183» 

and we find from the table of segments ^ = 93" 49' ; hence the 
radius of the cross-section of the pillar is 

z = r sin. 46° 50' = 0,729 r. 
To resist rupture by crushing, the radius of the cross-section 

n T 

always be employed for all points, where the formula for breaking 
across gives smaller values for z. 

If the pillar stands with its base unretained, as is represented 
in Fig. 437, the calculation must be made in the same manner for 

one-half ( - ) of it. The maximum radius r is, of course, that of 

the cross-section in the middle, and it corresponds to the formula 



of the pillar at the top must be r^ = y — ^, and this radius must 



p _ 3 



§ 268, Hodgkinson's Experiments. — The recent experi- 
ments of Mr. Hodgkinson upon the resistance of columns to 
breaking across (see Barlow's report in the " Philosophical Trans- 
actions," 1840) confirm, at least approximatively, the correctness 
of the formulas deduced m the foregoing pages. According to this 
experimenter the formula 

\2 // \2 // 64 \2 V 12 

for prismatical columns with circular or square cross-sections is 
correct for wood when we introduce a particular value for E\ but, 
on the contrary, it can be employed for wrought iron only when 
we substitute for d" the power 6?^''^, and for cast iron it is suffi- 
ciently correct when d'' and f are replaced by the powers d^'^^ and ?''\ 
The chief results of Hodgkinson's experiments upon prismatic 
pillars with circular and square cross-sections are given in the fol- 
lowing table. The coefficients giA^en in it refer to the case when 
the pillars are cut off at both ends at right angles to their longitu- 
dinal axis and repose upon these bases. When the ends are rounded 
so that these extremities of the columns are not prevented from 
assuming any inclination, these coefficients are nearly tliree times 
as small. If, on the contrary, the column is fixed at one end and 
the other capable of turning, the coefficient is hnt hcdfas gredt i\s^ 
in the first case. If, finally, one end of the pillar is fixed and the 



268.] PROOF STRENGTH OF LONG COLUMNS, ETC. 



543 



other capable of being turned and of sliding, the proof load is but 
one-tenth of that of the first case, where both ends are fixed. 



TABLE OF THE FORCES NECESSARY TO RUPTURE COLUMNS BY 

BREAKING THEM ACROSS. 



Name of the prismatic pillars. 


Breaking stress. 


English measure, 
tons. 


French measure, 
kilograms. 


Prussian measure, 
new pounds. 


Cast-iron pillars with circu- 
lar cross-section . . . . 

Wrought-iron pillars with 
circular cross-section . 

1 

Square pillars of dry Dantzic 
oak 

Square pillars of dry fir . . 


44,i6 ^,,, 


10900 p,, 
46140 ^, 

2480— 

b' 
i770y 


94700 ^,^, 
^3,55 

284400— r^- 

b' 

23570^, 

16840^, 



In the column for English measure d and h are given in inches, 
I in feet, and P in tons of 2240 pounds. In that for the French 
measures, on the contrary, d and b are given in centimetres, I in 
decimetres, and P in kilograms, and in the last column d and b are 
expressed in inches, I in feet, and P in new pounds. 

Mr. Hodgkinson also found that cast-iron pillars, with round 
ends, were sooner crushed than broken across, when I <^lb d, and 
when the ends were flat as long as I was < 30 d. Dry wood possesses 
double as much strength as timber just felled. When employing 
this formula for calculating the working load of columns, we employ 
a coefficient of security of I to jV or a factor of safety of from 4 to 12. 

Hence, with sextuple security, we can put for cast-iron pillars, 
when d and I are given in inches. 



44 16 r7^'^^ 



44 1 6 d^'^^ d^'^^ 

' . 68,3 ^ = 502,688 ^ tons. 



6 



I 



V'' 



6 V'' 

and d = 0,0173 (P V^y^'"''' inches. 

For wrought-iron pillars we have, when we adopt the same 
coefificient of security, 

P = 3210 ^~ tons and 

d = 0,01028 (P P) »'"'' inches. 



544 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 269. 



For inllars of oak wood, employing a coefficient of security of 
j*^, we have 

P = 157,68 ( ^ )V = 157,68 y* = 267,69 ~ tons, 

b = 0,2822 (P f)^ and d = 0,2472 (P ly inches. 
Finally, for pillars of fir wood, we have 



P = 112,46 Ij\\f= 112,46 



^ 

f 



190,92 y. 



^> = 0,307 (P r)^ and d = 0,269 (P/^)i 
Example. — For a cylindrical fir post, 11 inches thick and 12 . 13 = 144 
inches long, fixed at both ends, the proof load is 

P = 190,92 ( ^4 ) = 134,802 tons. 

If the ends of such a pillar are capable of moving freely, the proof load 
P = ^ P = 44,984 tons, while according to the theoretical formula we 
have Pi = 56900 lbs. = 25,402 tons. (See Example 1 of § 266.) 

§ 269. More Simple Determination of the Proof Load 
of Columns. — The foregoing formulas for the bending and 
breaking across of pillars are calculated upon the assumption that 
the force P is applied exactly at the end A of the longitudinal axis 
of the pillar. Now since in practice this is scarcely ever perfectly 
true, and since the action of the force ceases to be central as soon 
as the pillar bends, it is advisable, in determining the proof load 
of a beam, to take into consideration from the beginning the 
eccentricity of the point of application of the force. Assuming 
that the point of application D of the force P is at a distance 
D A = c from the end A of the axis A P, Fig. 441, of the column 
and that the deflection B C = a oi the pillar is small, 
compared with c, we can consider the elastic curve 
formed by the axis of the pillar to be a circle, whose 
J" 



Fig. 441. 




radius is r = 



%d 



But now 



P {a -{■ c) r =^ WE, whence 

P (« + c) ^ = 2 TT P «, as well as 

P? c 



a = 



a + c = 



2 WPJ - PI' 
2 WBc 



and 



2 WPJ - PI' 

If F denotes the cross-section of the pillar and e half its thick- 
ness, measured in the plane A B D, the uniform strain produced 
in each cross-section by the force P is 



369.] 



PROOF STRENGTH OF LONG COLUMNS. ETC. 



545 



s, 



p 



and the strain produced at the exterior surface by the moment 

P {a + c) of the force is 

P (a + c )e _ 2 PEce 

~ 2 WE- pr 



So = 



w 



and consequently the maximum strain in the pillar is 

P 2 P Ece _ P /. % E Fee 

P ^ 2 WE- P I' ~ F \ ^2 }VE-P P 



S =^ Si -{- jS^ 



p = 



p = 



Putting S = to the modulus proof strength T, we have 
„/^ 2EFce \ ^^ 

P(2 WE - PI' -^ 2E Fee) = {2 WE- P ^) F T. 
Now if P f is small compared with {W + Fee), we can put 
2 WEFT FT 



2 EiW + Fee)'+ FTt" , Fee FT^ 



, or 



W 2 WE 



FT 



+ V 



, in which and i/^ are empirical numbers. 



The civil engineer Love (see " Memoire sur la Eesistance du fer 

etdelafonte, etc.," Paris, 1852) deduced from the experiments of 

Hodgkinson the values = 0,45 and ip = 0,00337; hence we have 

FT 
P = xFT = 



1,45 + 0,00337 /J V 



from which the following table for the coefficient 

■ ^ = 1,45 + 0,io337 7F' ^"^ '^^'^ calculated. 



©' 



I 

d~ 

X= 


lO 


20 

o,357 


30 
0,223 


40 


50 
0,101 


60 


70 


80 
0,0435 


90 
0,0347 


100 


o,559 


0,146 


0,0735 


0,0556 


0,0285 



These values of x must be multiplied by the modulus of proof 
strength T for compression, when the modulus of proof strength 
for long pillars is to be determined for a given ratio of length. 

General Morin gives, after Rondelet, the following table, which 
35 



540 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 269. 



furnishes too great values for %, when the pillars are of medium 
length. 



I 

d 


I 


12 


24 


2>^ 


48 


60 


72 


X = 


I 


5 
6 


1 

2 


1 

3 


1 
6 


1 
12' 


1 

•2 4 



Example — 1) What load can a pine post bear, whose length is 15 feet 

and whose thickness is 13 inches ? According to the table upon page 404, 

the modulus of proof strength for a short pillar is T = 2600 ; but since the 

I 
ratio of the length to the thickness is ^ = V-) "^^ have 

V — — = 453 

^ - 1,45 + 0,00337 . 15^ 3,208 ' ' 

whence we obtain the modulus of proof strength x T = 0,453 . 3600 = 

1178 pounds ; hence the proof strength of the pillar is 

P=z 1178 -J- = 1178 . 0,7854 . 144 = 133000 pounds. 

If we employ a factor of safety 3, we can put 

133000 ,,^^^ , 

P = -^ — = 44300 pounds. 
o 

3) How thick must a hollow cylindrical pillar of cast iron, 25 feet 
long, be made, when it stands vertical and is required to support a 
load P = 100000 pounds ? Assuming the diameter d^ of the hollow part 
to be three-fifths of the exterior diameter {d) of the pillar, we can substi- 
tute in the theoretical formula 



P = 



= ^ . f ^(§236), 



^4 ^ 



d 



d' 


16 


11^ 


d' 
16 


4 
4/ 


/ 


4.PI 


2 



[1 _ (sy-j — 0,0544 d\ whence we obtain 



y 0,0544 Tt^^ 

Substituting in this expression P = 100000, l^ = (35 . 13)' = 90000, 
TT^ = 31, and, instead of jE7, 
E 

To 



14220000 ^ ^^^^^^^ 



10 
we obtain the required thickness of the pillar 

^ = "^^ 0,0544 . 31 . 1433 "" I 
4 






6000000 



187500 



1,6864 . 337 
= 11,07 inches. 



0,0537 . 337 
If we make d = 11,35 inches, we obtain d^ = 0,6 . 11,35 = 6,75 inches. 



I 



§270.] COMBINED ELASTICITY AND STRENGTH. 547 

According to our last formula we have, when we assume 

for the required cross-section of the pillar 

3,556 . 100000 355600 



r a \n P 3,556 . l< 

F = [1,45 + 0,00837 y-^ J ^ = -^ Y 

and putting, according to § 212, 

^ 18700 ^^^^ 
T = ^- = 6200 pounds, 
o 

we obtain 

355600 -, , . 

F = = 57,35, and therefore, since 

i^ = -^ (d' -d,^) = \\- im -^- = 0,16 TT ^% 

the required exterior diameter of the pillar 

^ = i^l^r. = / W- = "'<^8 iBches. 
Assuming d = 11 inches, we obtain 
d^ = 0,Q d = 0,Q . 11 = 6,6 inches. 



CHAPTER y. 

COMBINED ELASTICITY AND STRENGTH. 

§ 270. Combined Elasticity and Strength. — A body is 
often acted upon at tlie same time by two forces, e.g. a tensile and 
a bending one, etc., by wiiicli a double change of form is produced, 
as, E. G., an extension and a bending. We call the force with which 
a body resists this two-fold change of form its combined elasticity 
and strength, and w^e will proceed to investigate the most important 
cases of this kind. 

Properly speaking, the case (§ 214) of the bending of a body 
A K B 0, Fig. 442, is really one of combined strength ; for the 

force A F = P, which acts at the end A of the body, can be re- 
solved into a couple (P, — P) and a force ^^"P = P. The former, 
which alone we have previously considered, tends to bend the por- 
tion A S of the body, and the latter tends to tear this piece from 



548 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 370. 



the remaining portion S B, The latter force can be resolved into 

two components 

Fi = F cos, a 



Fig. 442. 




and 



Fi = F sin, a 



(§ 215), one of which 
acts at right angles to 
the direction of the fibres 
and the other in the di- 
rection of the axis of the 
fibres. The latter com- 
ponent combines with 
the strain in the fibres 
produced by the bend- 
ing and increases the ex- 
tensions upon the side of 
the tensile strains and 
decreases the compres- 
sion upon the other side. 
The magnitude of the 
extension of each fibre 

RS = K N, 

etc., whose length = 1, 
by the tensile force F 
sin. a is (§ 204) 



F E 



F denoting the cross-section N oi the body. 

If at this distance from the line Ny Oi, Fig. 443, which deter- 
mines the ends of the fibres, that have been extended by the bend- 
ing, we draw a line iVs ^2 parallel to it, it will form the boundary 
of the fibres which have been submitted to both causes of change 
of length, and it will cut the original limit in a point /S'o, which 
corresponds to the fibre, that is unchanged in length, and conse- 
quently gives the new or true position of the neutral axis. The 
distance 8 S^ = Si of this neutral axis from the original one, which 
corresponds to the moment of flexure, is determined by the pro- 
portion 



270.] 



COMBINED ELASTICITY AND STRENGTH. 



549 




S Si 
S Si 



SJV ^ _^ 

NM,'^-^-' e ~ a' 



whence ei = - ^i. 

G 

But we have also - = - (8 235), 
e r 

hence 



P r sin. a 



F E 

The radius of curvature r^ of 
the neutral axis determined in 
this more accurate manner is 
greater by the quantity (ei) than 
that of the neutral axis previously considered ; hence we have 

ri = r + ei = r (1 + <7i) =: r ^1 + —Ye)' 

The angle a, which the variable cross-section JV, 0, or iVg 0^ 
forms with the direction of the force P, is equal to the tangential 
angle a (found in § 216) ; hence, as this angle is small, we can put 

P {f - x') 



or, since 



stn. a = a 
WE 



P X 



2 WE 
(§ ^15), 



V — x^ 
T sin» a = r a ^::z , from which we obtain 



ex ■= 



2 x 
P {f - 



x^) 



% F Ex 

Hence for the point B, where the beam is fixed and for which 

a; = ?, we have e^ = 0, and for the point A at the other end, where 

Pf ,, , . P(l' - x") 

X = Q, e^ = —zr-- = 00 ; on the contrary, for x = ' „ ^ — - we 

'0 ' 2 F E e 

have e, =: e ; consequently the neutral axis coincides at B with 
the original one, and in passing from 5 to ^ it separates more and 
more from it, until, finally, it reaches the concave side of the body, 
and, if prolonged beyond the body, at the end A it is at an infinite 
distance from the other axis. 

The maximum extension produced by the flexure is 

Pex 



a = 



WE' 



550 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 370. 



and that produced by the tensile force P sin, a is 



P sin. a 



o, = — 



FU ' 



hence the total extension is 



e X sm. a^ 



and, if the latter has reached the limit of elasticity -^, we can put 

/e^ sin. a \ _ 



and the proof load is 



P --= 



W T 



W T 



e X -{- -=, sm. a 
F 



p (r - x')' 



e X + 



2 F E 

For a moderate deflection, which is all these girders are gene- 
rally exposed to, this value is a minimum for x ^ l, and it is 

WT 



P = 



el ' 



as we have already found. 
Fig. 444. 




Remark. — If the girder, as, e.g., A A^ jB, 
Fig. 444, I., II., III., is acted upon by two 
forces, two or even three displacements of 
the neutral axis from the centre of gravity 
may take place. If the two forces act in 
the same direction as represented in Fig. 
444, I., this displacement on one side of the 
cross-section A^ is determined by the 
formula 



e^ -= 



P r sin. a 

'Ye"' 



and, on the contrary, on the other side by 
the formula 

(P + Pj) r sin. a 



e„ = 



F E 



At the point of application A^ this dis- 
placement changes from 

P r sin. a 



A^ V, = e,= 



FE 
P + P 



to 



/P + P\ 

^1 T^2 = ^3 = \ P I ^" 



when we pass from one side to the other, 
on the contrary, at the fixed point P, where a = 0, we have e^ = 0. 



§ 371.] 



COMBINED ELASTICITY AND STRENGTH. 



551 



If the two forces act in opposite directions and the moment 

of the negative force is greater than the moment 

P . ATB = P{h + I) 
of the positive one, in which case the girder is bent in two opposite 
directions, which meet in a point of inflection F, the neutral axis consists 
of three branches U V^, V^ W^ and W^ B (Fig. 444, II.), which are not 
continuous, and the normals at the point of inflection F is an asymptote 
to the last two of these curves ; for here r = co and consequently 



«i = 



Pr sin. a 
~F^ 



= 00 



If, although the forces act in opposite directions, we have P(l + l^) > P^l^^ 
as represented in Fig. 444, III., the displacement of the neutral axis upon 
one side of -4 ^ is 

P r sin. a 



A,V,=e,= 



FE 



and that upon the other is 



^1 y2 = H = 



(P — Pi) r sin a 
F~F ' 



and at the cross-section through A^ there is a break in the two branches 
U Fj and Fg B of the neutral axis, the value of which is 



1^^ P^ r sin. a 



PcL^<J 



Fig. 445. 




Fig. 446. 



§ 271. Eccentric Pull and Thrust— If a column A B, Fig. 
445 and 446, acted upon by a tensile or compressive force, whose 
direction, although parallel to, is not that of the longitudinal axis 
of the body, the combined elasticity and strength will come into 
play. This eccentric force can, as we know, be replaced by a force 

P in the direction of the axis, 
and a couple (P, — P), whose 
lever arm c is the distance C A 
of the point of application of the 
force P from the axis of the body, 
and whose moment is therefore 
— P c. The force ^ P = P in 
the line of the axis produces in 
all the fibres the constant strain 

P 

Sx — -^, in which F denotes the 

cross-section of the body ; the 




552 GENERAL PRINCIPLES OF MECHANICS. [§371, 

couple, on the contrary, bends the body in a curve, whose radius 

is determined by the well-known formula (§ 215) P x r = WE, 

in which we must substitute for the moment of the force the 

WE 
moment P c of the couple. Consequently r — -^ — is constant, 

when W or the cross-section F is constant, and therefore the curve 
formed by the axis of the body is an arc of a circle. 

If e is the maximum distance of the fibres from the neutral axis 
passing through the cross-section of the body, we have the maxi- 
mum strain produced in the body by the couple 

_P c e 

and hence the total strain is 

consequently, when we put this equal to the modulus of proof 
strength T, or assume that the most remote fibre is strained to the 
limit of elasticity, we obtain 

F W ~ \ W I F' 

Hence the proof load of the pillar is 

F T 



P = 



, Fee' 
1 + 



W 
E.G., for one with a rectangular cross-section, the dimensions of 

which are & and h, 

FT 
p — 

D C 

and for one with a circular cross-section, whose radius is r, 

FT 


4 c 
1 4- — 
r 

From this we see that the strength of a body is tried much 
more severely by an eccentric pull or thrust than by an equal one 
acting in the direction of the longitudinal axis of the body. 

If the column is prevented from bending by a support upon the 
side, as, e.g., BAG, Fig. 447, represents, P remains of course 
= FT 

If the force acts at the periphery of a parallelopipedical pillar 

h 
A B, Fig. 448, and at the distance c — - from the axis, we have 



§ 272.] 



COMBINED ELASTICITY AND STRENGTH. 



553 



^ 



P = 



FT 



= \FT% 



Fig. 447. 



Fig. 448. Fig. 449. 



1+3 

and the proof load is but one-fourth of what it would be if the 
weight were applied in the prolongation of the axis of the body 
(Fig. 449). 

For a cylindrical pillar, with 
a force acting at the circum- 
ference, we have c — r, and 
consequently 

F T 
P = -^^- ^\FT, 

1 + 4 ^ ' 

I.E., but one-fifth what it would 
be if its point of application was 
in the axis of the body. 

These formulas can be applied 
to rupture by extension, com- 
pression and breaking across ; it 
is only necessary for each species 
substitute a different coefficient of ultimate 





of separation to 
strength, or put 



P = 



F K 



F 



F c e 

w 



+ 



Fee' 



in which K^ denotes the modulus of rupture by compression (or 
extension) and A^, that for breaking across. 

"^ § 272. Oblique Pull or Thrust. — The theory of combined 
elasticity and strength is particularly applicable to the case, where 
the direction of the force P forms an acute angle R A P ^ d with 
the axis of the beam A B, Fig. 450. One of the two components 

R z=z P COS. d acts as a tensile force 
and the other P sin. 6 as a bending 
one upon the body, and the strain 
^ _ P COS. 6 

produced in the whole cross-section 
by the first component combines with 
the strain 
P sin. 6 . I e 



Fig. 450. 




So = 



produced by the moment P I sin. d of the second component in 
the outside fibres, and causes the strain 



554 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 272, 



m CI CI , a P ^os. 6 Fie sin. 6 
I = O = Oi + 02 = ^ — + 



or more simply 



T 



'H 



COS. 6 



+ 



F 
I e sin. 6 



W 



F ' W 
Hence the required proof load is . 

p_ ^Z 

COS. o -\ — :riT- sm. o 



)• 



W 
or, inversely, the required cross-section is 

F=^ [cos. d , ^ 

Or, if we substitute a modulus of proof strength T^ for bending 
different from that (T) for extension we have 



Fie . A 
- stn. o\. 



F 



'"{ 



cos. d Fie 



. (j). 



Por a parallelopipedical girder we have 
-^ = p and consequently 



F^ P i^ 

and for a cylindrical one 



COS. 6 6 Z 



h T, 



J- sin. SV 



Fe 4 , 
-fj^ = -, whence 
W r 



F 



"■( 



COS. S 4:1 

+ 



f sin. dj. 



Fig. 451. 



T ' r Z 

The same formula holds good for the case represented in Fig. 
451, in which the first component R produces compression in the 

girder. If here again d denotes 
the angle, which the direction 
l !li!^):)!l!','j!!l!!! ' i!a ! of the force P makes with the 
axis of the girder, the values of 
the components are 

E = P COS. S and 
N zzz P sin. 6. 
In order to find the proof 
load of the girder, we must com- 




bine the strain produced by M 



S^ = 



P COS. 6 
~F 



§:^72.] COMBINED ELASTICITY AND STRENGTH. 555 

with the greatest strain 

^ _ P ? e sin. 6 , 

produced by the bending, and then we must substitute in the 

formula 

rr, T^ ( COS. 6 le sin. d\ 

^ P I . FU . A 

■F=Y v^^' ^ 'W ^'^" 7 

just found, instead of T, not the modulus of proof strength for ex- 
tension, but that for compression. 

In both the cases treated above the displacement of the neutral 
layer of fibres from the centre of gravity is 

_ (7i _ ^S'l _ W cotg. 6 

B\ — 6 — "77" 6 — ^ • 

(T2 8<i F ex 

which, E.G., for parallelopipedical beams, becomes 

_ li cotg. 6 

It is also easy to see that by the combination of the maximum 
extension or compression with the extension or compression of the 
fibres, which is equally distributed over the entire cross-section of 
the body, there is produced an extension or compression 

Si ± S2 P (cos. 6 I e sin. 6] 



(Ti ± (72 = 



_ P I COS. o I e sin. 6\ 



E 

If we introduce the modulus of proof strength T and for the 

T 
sake of security employ for wood and iron only — , we obtain 

1) for wood in doth cases 

„ ^ 780 P 1^^_ 

COS. o + -y- sm. cos. + — sm* 
h r 

2) for cast iron, in the first case (Fig. 450) 

3640 F 3640 F 



P = 



COS. + -=~ sm. o COS. + — sm, o 
h r 



and in the second case (Fig. 451) 

9360 F 9360 F 



P = 



A , 6^ . . A . ^^ • A 

COS. o 4- -7- stn. COS. OH sm, o 

h r 



?>56 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 273. 



§ 273. The case just treated occurs often in practice. If, e.g., 
a weight P is hung from a girder A B, Fig. 452, which is inchned 
to the horizon, we have, when the angle of inchnation of the direc- 
tion of the axis h P A R = 6, the tensile force R =^ P cos. d and 
the bending force N — P sin. 6, and therefore 

FT 



P = 



COS. o -i- -Y- sm. o 
a 



Fig. 453. 



Fig, 453. 





If, as is represented in Fig. 453, not only the direction of the 
stress P is inclined to the axis of the body, but also its point of 
application lies without it, in calculating the proof load we must 
consider the point of application transported to D in the pro- 
longation of the axis A B of the girder, i.e. we must substitute in 
place of the length B A — I the length B D = B A + A D = I -\- 



sm. 



— 1> in which the horizontal distance G A h denoted by c, and 

the angle D A, formed by the axis of the girder with the verti- 
cal, is represented by S. 

In like manner, for the pillar A B, Fig. 454, which is inclined 
at an angle 6 to the vertical, we have the proof load 

FT FT 



P 



COS. o -{- -J- sm. 



4:1 . ' 

COS. (5 H sin. 6 

r 



in which we must substitute the modulus of proof strength for 
compression, while in the former case we should employ that for 
extension. 

If a loaded girder A A, Fig. 455, is not freely supported, but 
wedged between two walls, a decomposition of the forces takes 
place into components producing compression and into compo- 
nents producing a flexure. If the terminal surfaces A, A of this 



373.] 



COMBINED ELASTICITY AND STRENGTH. 



557 



beam form an angle 6 with its cross-section, and if a force P acts 
in the middle B of the girder, the reactions of the walls upon the 
ends of the girder are Q and Q, and these forces are inclined at an 



Fig. 454. 



Fm. 455. 




angle S to the horizon and give a resultant C P = — P, which 
balances the force P. 
Hence 

P = 2 Q COS. A CP = 2 Q sin. 6, 
or inversely 

2 sin. d 
The reactions of the walls can be decomposed into a compres- 
sive force in the direction of the axis of the girder 

P cos ^ 
R = Q cos. <^ = -TT -^-— t: = i P cotq. 6 
^ 2 sm. 6 ^ ^ 

and into a force 



N — Q sin. 6 — 



2' 



which is perpendicular to the latter and produces a bending ; con- 
sequently we have 



I.E. 



T = 8 =^ 8i + /S'a = -^ + 



T= ^ !^^£' ^ I ^^^ 



w 



2 F ' 4:W' 
and the proof load of the girder is 

2 FT 



P ^ 



cotg. <J + 2 



Fie 
W 



The condition of affairs is the same, when an inclined prop A B, 
Pig. 456, carries a load which has been dumped upon it. But here 
Q can be resolved into a force Qi at right angles to the axis of the 



558 



GENERAL PRINCIPLES OF MECHANICS 



[§ 273. 



Fig. 456. 



prop and into a force iVj at right angles to the side (in miners' 
language, the floor). Neglecting, for greater safety, the friction of 

the loose masses of stone upon 
the floor and denoting the angle 
formed by the terminal surfaces 
of the prop with its cross-section 
by (5, and the inclination of the 
floor B C to the horizon by (3, we 
obtain Q^ = Q sin. (3 and 

2FT 




cotg. ^ + \ 



FJe 
W 



(see § 240), and therefore 



I cotg. ^ + ]- — TTT-j sin. /3. 



Example — 1) "What must be the dimensions of the cross-section of the 
inclined girder A B, Fig. 452, which is made of pine and is 9 feet long and 
whose direction forms an angle of 60° with the horizon, when it bears at 
the extremity A a weight P = 6000 pounds? The formula 

FT 



P = 



COS. 6 + -,— Sin. 



gives, when we substitute P = 6000 pounds, T = 780, d 
30° and Z=9.12 = 108 inches, and assume r = f , 



90° - 60° = 



F=dh = ^h^ = 



6000 

780 



I COS. 



30° + 



6. 108 



h 



- sin. 30 



'), I.E. 



/ 648 . 0,500\ 3489 

A^ = 10,77 ( 0,866 + j^ — ) = 9,33 + -^. 

Approxiraatively, we have 

h = VUSQ = 15,17, 
more accurately 

h = V3489 + 9,33~.lV'7 = ^3631 = 15,37 inches, 
and consequently 

& = -5 A = 10,98 inches. 

2) At what distance from each other must two 12 inches thick collars 
J. jB of a so-called overhand stoping A B G, Fig. 456, be laid, when the 
gob is T)iied 60 feet hisfh upon it in a vein 4 feet thick, dipping at 70°, if 
we affu .1'^ that the weight of the gob is 65 pounds per cubic foot? De- 
noting the required distance by x, we have the weight upon each collar 



§274] COMB-NED ELASTICITY AND STRENGTH. 559 

§ = 4 . 60 . 65 a; = 15600 aj, and consequently the pressure upon each 
collar is 

Q^ = Q sin. 70° = 15600 x sin, 70" = 15600 . 0,9397 x = 14659 x lbs. 
If the ends A A of the collar form an angle of 70° with the axis, or if' 
iJ = 20°, we have 

2FT 2.113,1.780 176436 



14659 X = 



3Z ~ „„_ 2.48 
cotg, 20° + -J 3,747 + -j- 



10.747 ' 



and therefore 



X = 



176436 



1,12 feet = 13,44 inches. 



FM 




10,747 . 14659 

The required distance between the two collars is therefore 
t cc — <Z = 1,44 inches. 

(§ 274.) Flexure of Girders Subjected to a Tensile 
I'orce. — The normal proof load P of a girder A B, Fig. 457, is dimin- 
ished by the application of a small force in the direction of the axis 
only when the girder is short. If, on the contrary, the length of the 
Pj^ ^g^ girder and the tensile 

force exceed certain 
limits, the moment of 
the latter acts in the 
opposite direction to 
the moment of the 
bending stress, thus di- 
^ minishing the deflec- 

tion of the body and increasing its proof, load. 

If we put again the co-ordinates of the elastic curve A S B, 
Fig. 457, formed by the axis of the girder, A K — x and K 8 — y, 
we have the moment of the forces in reference to a point S in the 
axis P X — Q y, 

we can therefore write (according to § 215) 

{Px- Qy)r= WE, 

substituting 

__ dx 

in which a denotes the tangential angle 8 T K, and denoting, in 

order to simplify the expression, y -j^^^ by p, and y -^^^ by q, 
obtain the equation 



WU 



WE 



we 



da — — 



dx 



{P x — Q y) d X . „ a \ -f 

- ^^ ^^j^ = _ (/ a; - q'y) dx. 



560 GENERAL PRINCIPLES OF MECHANICS. 

Now making 



[§ 274. 



'p'^ X 



1) y --^- (^«'* + ne-''% 



Fig. 458. 




i 

W/a 



in which m and n de- 
note constants, to be 
determined, and e the 
base of the Naperian 
system of logarithms 
(see Introduction to the 
Calculus, Art. 19), we 
obtain 



2) 



a 



_ ^y __ jp^ 



(m e'^'' — n e~* *) q. 



dx (f 
and since the differential of the last equation, viz., 

d a = — {m £'''' + n e-'i'') q^ d X, 
when substituted in equation 1), gives the above fundamental 
formula 



da = (^y -^^^q' dx= -^ {p'^x - q-'y) 



d X, 



the correctness of the above expression for y is proved. 

Since for a: = we have ^ = 0, we obtain by substituting these 
values in 1) the following equation 

= — (m e° + 7^ e"), I.E., 

m + n — 0, 
and since for a; = /, a = 0, we obtain by substituting these values 
in 2) the equation 



= 



(m s'^^ — n e~^ ^) q, 



and substituting the value n = — m taken from the foregoing 
equation, we have 

= ^ — mq (e^^ + £"«% 
whence 



m = 



n = 



q' (e?^ + £-«') 
and the moment of the forces is 

Fx — Qy= Qm (e«" - e"'") 

p /£?» _ e-9n 

The latter is certainly a maximum for the fixed point B, the 
co-ordinates of which are x = A C = I and y = B C = a, and 
then its value is 



§275.] COMBINED ELASTICITY AND STRENGTH. 561 

K 2^ Hs a proper fraction, that is, if the girder is short and the 
force in the direction of the axis is small, we can put 



ei^ = 1 + ql ^ -^ + ^ + ..., 
and also 

hence we have the moment of the forces 

If, on the contrary, the force Q is so great that q I becomes at 
least = 2, we can then neglect 



g-? ^ — _ 
- g,/j 

when it occurs with e*^ ', and therefore we can put 

g?' + £-9' — ^ — » 

80 that the moment of the forces becomes simply 

^ q ^ Q 

(§ 275.) Proof Load of a Girder Subjected to a Ten- 
sile Force. — By the aid of the moments of the forces P and Q, 
found in the foregoing paragraph, we can determine by the method, 
which we have so often employed, the proof load of the girder. 

The force Q produces a tension per unit of s'lrface 

in the direction of the axis of the body, and the moment PI — Qa 
of the two forces P and Q produces a tension in the fibres at the 
maximum distance e from the neutral axis, which is 

^ _ (Pl-Qa)e 

hence the total tension is 

cr_c,,o_0 , (PI - Qa)e 
,b - 6, + 6, = -^ + ~ -. 

36 



563 GENERAL PRINCIPLES OF MECHANICS. [§375. 

When the latter reaches the limit of elasticity, 8 = T, and we 

can put 

£ {Pl-Qa)e 

^ - F '^ W 

If the modulus of proof strength T^ for compression is different 

from that 2^ for extension, we have 

T - _ i 4- iZJ-QAl 

^'- F ^ ~ W 
in which e denotes the maximum distance of the compressed fibres 
from the neutral axis. In both cases we must substitute 

Pl-Qa = -~ (~ — -—-), 

^ q \e''' -{- £-^7 

so that the required proof load of the body becomes either 

_ / e'^^+ e-^\ I _ _e_\ WT q 
^-\e^'-e-^7\ ft) e ' 



or 



^ - \e9'- e-^7 \ '^ F tJ e ' 
For a small tensile force Q we can put 

SO that, when we take into consideration the extension only, we 
have 

(FT-Q) w _i qr \ ( _ _Q_\ WT 

Without the tensile force Q the proof load of the body would be 

_ WT 

^'- le~' 
hence we have the ratio 

P 

Pa 

from which it is easy to see, that the proof load is increased or 

or 

diminished by Q, as ^7p is greater or less than ^^ „ , I.E., as 

3 W . T 

-^^Pj^ IS greater or less than -=^. 

When the tensile force is great, in which case we can put 
Pl-qa=^P \/~; 



', \ ^ %W EJV FTP 



§376.] COMBINED ELASTICITY A^^D STRENGTH. 563 

we have the proof load 

^ \ FT! ^ E e 
This expression becomes a maximum with the expression 



V Q — -^7-^. By differentiating the latter and putting the differ- 

ential equation obtained equal to zero, we obtain 

FT 

This maximum yalue is 



_oJF^yT T 



and the ratio of the latter to the proof load Pj of a girder, which 
is not subjected to a tensile force, is 

P,~ '^ ^ WE ~ '^ ^ 3 W 
For a parallelopipedical beam, whose height is h and whose 

width is by we have F = b h, W = -^^ and e = ^ h, whence 

P, 3h ^ E 3 1i 
If the beam is of wood, 

_ r _ J. 

"^ ~ E ~ 600' 
and therefore 

J; = i ^m ■ I = 0'«^^ I' 

E.a, for ,- = 30, P = 1,632 P, ; 

the girder carries nearly tico-tliirds more than when it is not sub- 
jected to a tensile force. 

For ^^ = --^- = 18,4, P, = P, and for values of ^ smaller 

than 18,4, P, is smaller than P, and the proof load P of the beam 
is diminished by the stress Q. 

§ 276. Torsion Combined with a Tensile or Com^ 
pressive Force.— If a column A B, Fig. 459, is acted upon at 
the same time by 2i force Q, whose direction is that of its axis, and 



564- 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 276. 



by a couple (P, — P), which tends to twist it, both the elasticity 
of torsion and that of extension (or compression) come into play. 
The result of the combination of these two elasticities may be in- 
vestigated as follows : If the strain per unit of surface produced 

by the force Q\^ Si = —^ and that produced by the moment of 

torsion at the distance e from the longitudinal axis of the body is 

P a e 
Si = — ^-, we can assume, that a parallelopipedical element 

Fig. 459. 



Fig. 460. 





A B C D, Fig. 460, of the body, is acted upon by the normal forces 

A B . Si and — C D . S^ upon A B and G D and by the couple 

(A B . Si, — C D . So) along A B and C D and by the opposite 

couple (B O , Z, — A D . Z) along B C and A D. If the diagonal 
plane A C forms an angle V' with the axis Of the body or with the 
direction of the strain Si, the components of the forces ^S'l, Sc^ and 
Z upon one side oi A C are 

A B . Si sift, tp, A B . Si COS. ip, and B C . Z sin. ip, 
and consequently the total normal force upon A Cis 

A G . S = ^^ . Si sin. ip + A B . S^ cos. ip + B G . Z sin. ip, 
or, since the moment of (^ G . Z, — A D . Z) is equal to the mo- 
ment of (A B . Si, — G D . Si), I.E. 

A B . B G . Z = B G . A B . Si or Z = Si, 

ATG". S = AB~. Si sin. ip + (TB cos. ip + BVsin. tp) Si, 
80 that, finally, the normal strain upon the unit of surface of 

A Cis 



§276.] C0MBI::ED elasticity and strength. 565 

^ ^ AB ^ . , , (AB , B C . \ ^ 

But -i-77 = sin. lb and -r-77 = cos. fp, whence 

AC ^ A C 

S = Si (sin. ip^) + 2 82 sin. ip cos. ip = Si (sin. ipy + S<i sin. 2 ip 

= Sil — — -^ — -] + S.2 sin. 2 \j) (compare § 259). 
This equation gives a maximum value for S, when tang. 2 ip = 
P9— or sm. 2-6 = and cos. 2w— —=:==z^ 

Si vs;' + {2S,f VSi'+{2S,f 

and this maximum yahie is 



oi /^ S\ \ 2 Si 



(1 + -^=^-=) 

\ Vs' + (2 s.)y 



s„. = ^1 + ^' + 



2 



^ \ V'Si' + (2 s^Y^ vsi" + (2 s^y 

Si 



IV(|)V.%'- 



Substituting the above vakies for Si and S^ in this equation, 
we obtain the required maximum strain 

Now, since the body should resist with safety the actions of 
these forces P and Q, we must put S^ = to the modulus of proof 
strength Tor 

2F ^ ^ \2FJ ^\W) ' 

from which we obtain the equation of condition 

QT 



/Pa e\ 



2 

-I = r 



F ' 

The allowable moment of torsion is therefore 



1) Pa = — \/t' - 



QT 



e ' F ' 

and the allowable force in the direction of the axis is 



^)Q = P'r-^(^-^) 



In order to find the dimensions of the cross-section correspond- 
ing to the forces P and Q, we put 
W ^ Pa. 

^ F 

when the force producing torsior is the greater, and, on the 
contrary. 



566 GENERAL PRINCIPLES OF MECHANICS. [§276c 

2 



F = 



T \ W I 



when that in the direction of the axis is the greater. 

For a parallelopipedical column, whose dimensions are h and h, 
we have 

F^hhW={l)'' ■]- h') -r| and e = i V¥~Th% consequently 



W^lh^;^^—T-. Pa 

e ~ Q 



/ f) T T \ h T J 



^ oh 



and 



rr _ z X _ Q - £ fl - ( ^-^^ \'1-' 

n. 36 IPaV tV \^WVif'lhT)\' 



{If + h') T 






If we know the ratio i* = - of the dimensions, we can calculate 

li 

the dimensions themselves by means of this formula. 

For a inllar with a square hase b = h, and therefore 

h'V2 _ P_a/ Q_\~^^ 

6 "TV h' TI ' 

i = » = ^i^i (i - jA)-, 

For a cyHndrical pillar or shaft we have 



Trr* 



F = -n r^^W — -^, and e — r, whence 
^^'_ P< ^ an<-\r-A/^^»l-i Q \~* ii 



nr 
( 






§ 277.] 



COMBINED ELASTICITY AND STRENGTH. 



5G7 



Fig. 461. 



If the force Q in the direction of the axis is a compressive one, 

the formulas found above still hold good ; for 
not only the direction of the force ^', (Fig. 
461) is opposite, but also the forces /S'o and Z 
can be assumed to act in the opposite direc- 
tion, when we wish to obtain the maximum 
resultant S,,,. 

Example. — If a vertical wooden shaft weigh- 
ing 10000 pounds is subjected to a moment of tor- 
sion P a = 73000, the required radius, assuming 




Q \- 



ix r'' T 



T = 400 pounds, is 

/ 7,958\-^ 

= V 0,6366 . 180 M — -^j . 

Approximatively, we have 

^ — . Vlii^ = 4,85, whence 
7,958 7,958 



W _ 3/0,6366 . 72000/ 10000 V 

/ ~~y 400 V ~ 400 77 Vy 



23,53 



3= 0,3383, and 



/, 7,958\-J 1 

(1 - -^ = -— ^ = 1,071, 

V r^ / V 0.6617 



V 0,6617 

so that the required radius is, more accurately, 

r = 4,85 . 1,071 = 5,194 mches, 
and consequently the diameter of the shaft is 

d = 10,39 inches. 

§ 277. Flexure and Torsion Combined. — Cases often oc^ 
cur where a girder or shaft is acted upon at the same time by a 
bending force and a twisting couple. Horizontal shafts are gen- 
erally submitted to both of these actions. In order to investigate 

the relations of the combined action of 
these two forces, let us imagine a pris- 
matic body A B C D, Fig. 462, fixed at 
one end B Z), to be acted upon at the 
other end by a bending force Q and 
at the same time by a twisting couple 
(P, - P). If I is the length A C oi 
the shaft, ir, the measure of the mo- 
ment of flexure and e, the maximum 
distance of an element of the cross-sec- 




568 GENERAL PRINCIPLES OF MECHANICS. [§277. 

tion from the neutral axis, we have the maximum strain produced 
in the direction of the axis by the force Q 

Sj = ^-^- (compare § 235). 

If, on the contrary, a denotes the lever arm H K oi the couple 
(P, — P), Wihe measure of the moment of torsion and e the 
greatest distance of any element of the cross-section from the axis 
C D oi the body, we can put the maximum shearing strain pro- 
duced by the couple 

I t 
Now here, as we can easily understand, the strain Sx = -^^- 

takes the place of the absolute strain S^ = ^^ of the foregoing par- 

agraph, and therefore we can put for the maximum strain in the 
whole body A B C D, Fig. 462, 



^. == I + /(§-/+ >^^ or 



2 TFi ^ \ 2 ITW \ W /' 
from which we obtain the equation of condition 
(Paey_ „, Q l.e,T 

The allowable moment of torsion is therefore 



I) Pa-~y I - ^^ _ ^ r i ^,^, 

ana the bending force is 
2) Q = v-^ \t' ~ (^^^) \ from which we obtain either 
W Pa 



ex 



, or 






For a square shaft 



W h' V2 . W, ¥ . 

— = —- — and — = ^, whence 

eh e, b 



§ 277.] 



COMBINED ELASTICITY AND STRENGTH. 



h' = —fr-- (1 - -f^) , and 
_ l/ I'VlPa / _ 6 Q lA-^ 



d6a 



as well as 



h' T 
6 1^ P aV-\-^ 






7i^ 7^ 

while, on the contrary, for a cylindrical shaft, 

W 
e 



7T r ^ Wx r: r^ ^ . 

—7^— and — = -r- : hence we can put 






as well as 



* n T \ -tr'Tj ' 

Very often it is not a couple, but a force P, acting eccentrically 
to the axis, which produces the torsion in the body B CD, Fig. 463. 

Since such a force can be decomposed into an 

equal central force C P — + P and into a 
couple {P, — P), whose leyer arm is the dis- 
tance C A between the axis G D of the body 
and the line of application of the force P, we 
have here a case of combined strength, al- 
though there is no other force Q ; for the 
twisting produced by the couple (P, — P), 
combines with the bending produced by the 
axial force + P. The above formulas can 
be employed directly for determining the 

thickness of such a body, when we substitute in them P I — QU, 
If, in addition to the eccentric force P, there is another Q, 

whose moment is Q A, we must substitute instead of P I, P I + Ql^. 




570 



GENERAL PRINCIPLES OF MECHANICS. 



[§278. 



Fig. 464. 



§ 278. Bending Forces in Different Planes.— If a girder 
or shaft B C, Fig.- 464, is acted upon by two bending forces Q, and 

§0, whose directions C, §iand C.2 Q^, 
although at right angles to the axis C^ B 
of the body, are not parallel to each 
other, the portion C^ B of the body will 
be bent by two couples ( §„ — Q^ and 
(fe — C2)? the resultant of which must 
be found, when we wish to determine 
the nature and magnitude of the bend- 
ing. If ^1 and U denote the arms of the 
forces §1 and Q^ in reference to the fixed 
point B, Qi li and Q.2 h are their mo- 
ments, and if a is the angle formed by the 
directions of the forces, when passing 
through the same point, we have, according to § 95, the moment 
of the resulting couple 

B c = V{Qi liY + (ft Sr"+^"(ft ^1) jQ-J^'cmT^, 
and for the angle (3, which the plane of this couple makes with that 
of the couple (ft, — Q\), 

V2 ^2 




sin. 13 = 



Be' 



In order to find the intensity and the plane of this couple 
{Bf — B), we can reduce the force Q^ from 62 to C^, combine the 



reduced force Q 



ft I 



k 



- by means of the parallelogram of forces 



with the force §, and thus determine the resultant B^ ; the pro- 
duct Bi l^ = B CIS the value of the moment of the resulting couple 
and the angle Q^ d B is the angle (3, which the plane of this couple 
forms with that of the couple {Qi, — Q^). This plane is of course 
that in which the body is bent, and by the aid of the moment Bi l^ 
= B c, just found, we obtain the maximum strain in the body 

Bee 



S 



w 



or, putting this equal to the modulus of proof strength T, we have 

— - = ^(ft ^0' + (ft i^y + 2 (ft ^t) (ft2 ^2) cos:^ 

e 
If a twisting couple (P, — P), whose moment is P a, also acts 
upon this body A B, the maximum strain becomes 



8^^T 



B e ex 



A/(Bee,V (BaeV 



§278.] COMBINED ELASTICITY AND STRENGTH. 57I 

in which Wi denotes the measure of the moment of flexure, W that 
of torsion, ei the greatest distance of any element of the body from 
the neutral axis and e that of any element from the longitudinal 
axis of the body at B. 

From the above we obtain 
/Paey_ , Rce.T 
\ W I ~ W 

= T'- [(ft lY + (ft ky+ 3 (ft ?0 (ft I) COS. a] ^?! 

By the aid of the formulas of the foregoing paragraph the 
required dimensions of the cross-section of the body can be found 
by substituting in them instead of Q I the sum Q^ k + Q^ k- 

If only one bending force ft acts upon the body and if at the 
same time it is acted upon by a single twisting force P instead of 
a couple (P. — P), this force P can be resolved into a twisting 
couple (P, — P) and a force P acting upon the axis, so that 
instead of §2 h we must substitute in the latter formula P I. 

Final Remark. — Although there is no portion of mechanics which has 
been the subject of so many experiments as the elasticity and strength of 
bodies, yet much remains to be investigated and many points are still 
uncertain. Experiments upon this subject have been made by Ardant, 
Banks, Barlow, Bevan, Brix, Busson, Burg, Duleau, Ebbels, Eytelwein, 
Finchan, Gerstner, Girard, Gauthey, Fairbaim and Hodgkinson, Lagerjhelm, 
Musschenbrock, Morveau, Navier, Rennie, Rondelet, Tredgold, Wertheim, 
etc. The older experiments are discussed at length in Eytelwein's " Hand- 
bucli der Statik fester Korper," Vol. II., and also in Gerstner's "Handbuch 
der Mechanik," Vol. I. A copious treatise on this subject by v. Burg is 
given in the 19th and 20th volumes of the Jahrbiicher des Polytechn. 
Instituts zu Wien. Theories which differ somewhat from those given in 
this work are also to be found in this treatise. The experiments of Brix 
and Lagerjhelm have already been mentioned (page 394}. New and very 
varied experiments upon the reacting strength of different kinds of stone 
by Brix are reported in the 32d year (1853) of the transactions of the 
" Verein zur Beforderung des Gewerbefleiszes in Preussen." A simple 
theory of flexure by Brix is to be found in the treatise " Elementare Berech- 
nuug des Widerstandes prismatischer Korper gegen die Biegung," which is 
printed separately from the transactions of the Preussischen Gewerbeve- 
reins. Wertheim's latest experiments upon elasticity have already been 
mentioned (page 396). An abstract of Hodgkinson's experiments is to 
be found in Moseley's "Mechanical Principles of Engineering and Archi- 
tecture." Hodgkinson's principal work, the title of which is " Experimen- 
tal Researches on the strength and other properties of cast iron, etc.," was 
published by John Weale in 1846. A French translation of it by Pirel 



572 GENERAL PRINCIPLES OF MECHANICS. [§278, 

appeared in Tome IX., 1855, of the " Annales des Fonts et Chaussees," and 
an abstract of it by Couche in Tome XX., 1855, of the " Annales des 
Mines." Tredgold has published a treatise upon the strength of cast iron 
and other metals. The following works are also recommended for study. 
Poncelet's "Introduction a la Mecanique Industrielle," Part I, Navier's 
Resume des Leeons sur I'application de la Mecanique, Part I., translated 
into German by Westphal under the title " Mechanik der Bankunst," to 
which work Poncelet has made some additions in his theory of the resist- 
ance of rigid bodies (see his Manual of Applied Mechanics, Vol. II., trans- 
lated Into German by Schnuse). We would also recommend particularly 
the " Resistance des Materiaux " (Leeons de Mecanique Pratique), by A. 
Morin, which has been much used in preparing this work. We may men- 
tion further the " Theorie der Holz-und Eisenconstructionen mit besonderer 
Riicksicht auf das Bauwesen," by George Rebhan, Vienna, 1856, the work 
of Moll and Reuleaux (already quoted in page 469) upon " die Festigeit 
der Materialien," a " Memoire sur la Resistance du Fer et de la Fonte, par 
G. H. Love, Paris, 1852," as well as Tate's work upon the strength of mate- 
rials as applied to tubular bridges, etc. The theory of combined elas- 
ticity and strength was first treated by the author in " der Zeitschrift fiir 
das gesammte Ingenieurwesen (dem Ingenieur), by Bornemann, etc.. Vol. I. 
In the first volume of the new series of this magazine (Civilingenieur. 
1854) the graphic representation of the relative strength is treated by Mr. 
Bornemann, and the results of the experiments made by Bornemann and 
by Lemarle are also given. 

The theory of elasticity and strength will be treated of again when we 
discuss the theory of oscillation and of impact. 

Mr. Fairbairn's Useful Information for Engineers, I. and II. Series, gives 
the results of many experiments upon the strength of wrought iron of dif- 
ferent forms, as well as upon stone, glass, etc. From a theoretical point 
of view, we can particularly recommend, " Legons sur la theorie mathe- 
matique de I'elasticite des corps solides," par Lame, " A Manual of Applied 
Mechanics," by W. J. Rankine, the " Cours de Mecanique appliquee," I. 
Partie, by Bresse, and the " Theorie de la resistance et de la flexion plane 
des solides," par Belanger. The treatise of Laissle and Schiiblen, " Ueber 
den Bau der Briickentrager," is a fair exponent of the state of science uj)on 
this question, when it was written, and is therefore to be recommended. 
Riihlmann's " Grundziige der Mechanik," 3. Auflage (1860), contains also 
a treatise upon the resistance of materials worth reading. 

The " Civilingenieur " and the " Zeitschrift des deutschen Ingenieur- 
vereins " contain several valuable treatises upon the theory of elasticity 
and strength, particularly those by Grashof, Schwedler, Winkler, etc., as 
well as several good translations from the French and English of Barlow, 
Bouniceau, Fairbairn, Love, etc. The results of many experiuients by Fair- 
bairn, Karmarsch, Schonemann, Volkers, etc., are also given in these journals. 



^i-Z>- 



FIFTH SECTION. 



DYNAMICS OF KIGID BODIES. 



C HAPTE R I. 



THEORY OF THE MOMENT OF INERTIA. 



Fig. 465. 



§ 279. Kinds of Motion.— The motion of a rigid body is 

either one of translation, or of rotation, or a combination of the two. 

In the motion of translation (Fr. mouvement de translation ; Ger. 

fortschreitende or progressive Bewegang) the spaces described 

simultaneously by the different parts of the 
body are parallel and equal to each other ; 
in the motion of rotation (Fr. mouvement 
de rotation; Ger. drehende or rotirende 
Bewegung), on the contrary, the parts of 
the body describe concentric arcs of circles 
about a certain line, called the axis of rota- 
tion (Fr. axe de rotation ; Ger. Umdre- 
hungsaxe). Every compound motion can 
/ 11: [I be considered as a motion of rotation around 

/ |]piv I ^ movable axis. The latter is either varia- 

& jM^jjii JjIq qj. constant. The piston D E and the 

piston-rod B F of sl pump or steam engine, 
Fig. 465, have a motion of translation, and 
the crank A C has a motion of rotation. 
The connecting rod A B has a compound 
motion ; for one of its extremities B has a 
motion of translation, while the other A 
has a motion of rotation. The axis of rota- 
tion of a cylinder, which is rolling, is con- 



D 



E 



674 GENERAL PRINCIPLES OF MECHANICS. [§280 

stanl, while that of the connecting rod ^ ^ is variable ; for its 
position IS determined by the intersection M of the perpendicular 
B K to direction C B of the axis of the piston-rod and of the pro- 
longation of the crank C A (see § 101). 

§ 280. Rectilinear Motion. — The laws of motion of a mate- 
rial point, discussed in § 82 and § 98, are directly applicable to a 
rectilinear motion of translation. The elements of the mass J/,, 
ifs? ^3? etc., of a body, moving with the acceleration j9, resist the 
motion, by virtue of their inertia, with the forces M-^p, M^p, M^p, 
etc. (§ 54), and since the motions of all these elements take place 
in parallel lines, the directions of these forces are also parallel ; the 
resultant of all these forces due to the inertia is equal to the sum 
M^p + M.p + M^p + . . . — (i)[fi + i/2 + Jfs + . . .)jt? = Mp, 
when M denotes the mass of the whole body, and the point of ap- 
plication of the resultant coincides with the centre of gravity. In 
order to set in motion a body, whose mass is M and whose weight 
IB G =^ M g and which in other respects is free to move, we re- 
quire a force 

P = Mp = —^-, 
9 
whose direction must pass through the centre of gravity S of the 

body. 

If, in consequence of the action of the force P, the velocity c is 

changed to the velocity v while the space s is described, the energy 

stored by the mass is (§ 72) 



^M = ^-~^)G = (i-k)a. 



3 / \ %g 

Example. — The motion of the piston and piston-rod of a pump, steam- 
engine, blowing-machine, etc., is variable ; at the beginning and end of its 
stroke the velocity is = 0, and near the middle of it it is a maximum. If 
the weight tf the piston and piston-rod = G^ and if the maximum velocity 
at the middle of its stroke = «5, the energy stored by them in the first half 
of the stroke and restored in the second half is 

1^0 = 800 pounds and « = 5 feet, we have 

L = 0,0155 . 5^ . 800 = 810 foot-pounds. 

Now if half the stroke of the piston is s = 4 feet, we have the mean 
force, which is necessary to produce the acceleration of the piston in the 
first half of the stroke and which the piston exerts in the second half, when 
it is retarded, 



§ 381.J 



THEORY OF THE MOMENT OF INERTIA. 



675 



8 



G = 



310 

— - = 77^ pounds. 



Fig. 466. 



V^ 




2gs' 

§ 281. Motion of Rotation. — If the motive force P of a 
body A B, Fig. 466, does not pass through its centre of gravity 8, 

the body turns around that point, and at the 
same time moves forward exactly as if the force 
acted directly at the point 8, as can be shown in 
the following manner. Let us let fall from the 
centre of gravity 8 a perpendicular 8 A upon 
the direction of the force and continue it in the 
other direction until the prolongation 8 B ib 
equal to the perpendicular 8 A, and let us sup- 
pose that two forces + i B and — I B, which 
balance each other and are parallel to B, are applied at B. The 
force -\- I B combines with half the force B acting in A and gives 
rise to the resultant 

applied at the centre of gravity, while, on the contrary, the force 
— IB forms with the other half (^ B) of the force B applied in A 
a, couple ; hence the force B, applied eccentrically, is equivalent to 
a force Bi = P, which is applied at the centre of gravity, and which 
moves this point and with it the body, and to a couple (^ P, — 
^ P), which causes the body to turn around its centre of gravity 8 
without producing a pressure upon it. The statical moment of 
this couple is 



Fig. 467. 



= ^B.8A-{-lB.8B^B.8A = Ba, 
or equal to the statical moment of the force P applied in A in 
reference to the centre of gravity 8] the resulting rotation weald 
therefore be the same if the centre of gravity ^S' were fixed and P 
alone were acting. 

If a body A B, Fig. 467, is compelled, 
by means of guides D E, Pi P,, to assume 
a motion of translation, the eccentric force 

A B =^ B produces the same effect upon 



the motion of the body as an equal force 
acting at the centre of gravity, and the 
couple (I P, — ^ P) is counteracted by 
the guides. If a is the eccentricity 8 A 
of the force P, or the distance of its direc- 
tion from the centre of gravity 8 of the 
body, and if h denotes the distance H K 




676 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 382. 



between the perpendiculars to the guides at the diagonally opposite 
points jPand G and (JV, — N) the couple, with which the body 
acts on the guides, we haye, by equating the moment of the 
couples (i P, - 1 P) and {N, - N), ■ 

Nh = P a, and therefore 



a 



m 



jsr^^p, 





Fig. 468. 




Fig. 469. 



If, finally, the body A B, Fig. 468, is prevented from moving 
forward by the fixed axis C, the eccentric force 

A P = P produces the same effect upon the 
rotation of the body about this axis (7 as a 
couple (^ P, — ^ P) with the arm 2 C A = 
2 C B = 2 a, or with the moment ^ P . 2 « — 

P a ; for the remaining central force C P^ = 
Pi = P is counteracted by the bearings of the 
axis (compare § 130). 

§ 282. Moment of Inertia. — During the rotation of a body 
A B, Fig. 469, about a fixed axis 0, all points ifi, i/g? etc., of it de- 
scribe equal angles at the centre Jfj C JVi 
= M^ C iVsj etc., = (f)\ which, when the 
radii C D^ =^ C D^, etc., ~ one (1) are 
equal, correspond to the same arc 

J 

Pi Pi = Pa P^, etc., = (/) = ^- -n. 

Since the velocity is determined by 
the quotient of the element </> of the space 
and the corresponding element r of the 
time, the angular velocity (Fr. vitesse an- 
gulaire, Ger. Winkelgeschwindigkeit), i.e. the velocity of those 
points of the body which are situated at a distance equal to the 
unit of length (e.g. a foot) from the axis of rotation, is therefore 
one and the same for the whole body, and its value is 

w = - 

T . 

and in like manner the angular acceleration, or the acceleration of 
the rotating body at the distance = unity from the axis of rota- 
tion, is the same for the whole body, and its value is 




Ct> 



K 



§383 J THEORY OF THE MOMENT OF INERTIA. 577 

6) denoting the increase of angular velocity in the element of 
time r. 

In order to find the spaces ^i, So? etc., the velocities Vj, v^, etc., 
and the accelerations pi, p^, etc., of the points M^, M^, etc., of the 
body, which are situated at the distances C M^ — r^, C M^ — r^, 
etc., from the axis of rotation C, we must multiply the angular 
space </), the angular velocity w, and the angular acceleration p by 
7*1, rs, etc. ; thus we obtain 

Si = (f> Ti, s^ = (j) 7-2, etc., 
Vi = 0) 7*1, V2 = 0) r^i, etc., and 
P\ = K, 7*1, pi = K 7-2, etc. 
K the whole mass M of the body is composed of the parts Mi, 
M^y etc., which are at distances equal to the radii r^, 7*2, etc., from 
the axis of rotation C, the forces with which these elements of the 
mass resist the rotation are 

P^ = Mipi = fc Ml 7*1, P2 = M22h = fc Mi rs, etc., 
and their moments are 

P^ri = K Ml Ti, P^T^ — K M^ r^, etc., 
and the moment necessary to cause the body to rotate with the 
angular acceleration k is 

Pa = kMi ri, + K M^r^^ -{- .,, 

= K {Ml ri' + M, r,' -j-M,rs' + .. .). 
In like manner (according to § 84) the energy stored by the 
elements Mi, i/s? etc., while they acquire the velocities Vi, v^, etc., is 
Ai = iMiVi' = ic,'Miri\ 
A^= ^M^ v^ = iG)' M, r,% etc., 
and therefore the worh done in communicating to the whole body 
the angular velocity oj is 

^l^Gi'{Miri' + M,r,^ -h M,r,' + ...). 
The force of and the energy stored by a body in rotation de- 
pends principally upon the sum of the products Mi ri + M^ r.^ + 
M^r^ + ... of the different elements Mi, M^, etc., of the mass and 
of the squares of the distances 7*1, 7*2, etc., from the axis of revolu- 
tion. This sum is called the moment of inertia (Fr. moment d'in- 
ertie, Ger. Tragheits-, Drehungs- or Massenmoment), and we will 
hereafter denote it by M r^ or W. Hence the moment of the force, 
by which the mass M = Mi + M^ + . . ., whose moment of 
inertia is 

W = Mr' = Ml ri' + M,r,' + .. ., 
37 



578 GENERAL PRINCIPLES OF MECHANICS. [§383. 

has imparted to it tlie angular acceleration k, is 

1) Pa = K Mr' = fc W, 

and, on the contrary, the work done in putting the mass if in ro- 
tation with the angular yelocity o) is 

2) P s = i G)' Mr' = ^(o'W. 

If the initial angular yelocity of the mass was e, the work done 
in increasing it to w is 

We can also determine from the work done and the initial ye- 
locity e the final yelocity w ; it is 

~2P~s 



0) = j/^a ^ 



w 



Example. — If the body A B, Fig. 469, moyable about a fixed axis G 
Mid in the beginning at rest, possesses a moment of inertia of 50 foot- 
pounds, and if it is set in rotation, by means of a rope j)assing round a 
pulley, by a force P = 20 pounds, which describes the space s = 5 feet, 
the angular velocity produced is 



./2 P s , /2 . 20 . 5 ,- 



1^ = r --^0— = V4 = 2 feet, 

I.E., every point at the distance of a foot from the axis of rotation de- 
scribes, after this work has been done, 2 feet in each second. The time of 
one revolution is 



2 7T 



t — — = 3,1416 seconds, 



G) 



and the number of revolutions in a minute is 

_ 60_ 60 _ 
"" ~ t ~ 3,1416 "" ■^^'^• 
If the angular velocity w = 2 feet, just found, is transformed into a ve- 
locity e = f foot, the work performed by the body is 

P^ Si = [2- - (|)^] . -V- = (4 - -jS^) . 25 = ff . 25 = 85,93 foot-pounds, 
E.G., it has lifted a weight of 10 pounds 8,593 feet high. 

§ 283. Reduction of the Mass. — If the angular velocities of 
two masses M^ and M.2 are the same, if, e.g., they belong to the 
same rotating body, their living forces are to each other as their 
moments of inertia TFi =■ M^ r^ and Wo = M.^ r^, and if the latter 
are equal, both masses have the same living force. Two masses 
haye, then, equal influence upon the state of motion of a rotating 
body, and one can be replaced by the other, without causing a 
change in that state, when their moments of inertia J/, r^^ and 
Mi r^ are equal, or when the masses themselves are to each other 
inyersely as the square of their distances from the axis of rotation. 



283.] 



THEORY OF THE MOMENT OF INERTIA. 



579 



With the aid of the formula Mx Vx = M^ r^ we can reduce a mass 
from one distance to another, i.E. we can find a mass M^, which at 
the distance r^ has the same influence on the state of motion of the 
rotating body as the given mass M^ at the distance r^-, and this 



mass IS 



M,^ 



Mx Tx' 



Tfi 



•2 ? 



I.E., the mass reduced to the distance r<i is equal to the moment of 
inertia of the mass divided hy the square of that distance. 

Two weights Q and Qx, fixed upon a disc A C B, Fig. 470, at 

the distances C B = h and C Bx = a from 
the axis of rotation XX, have the same 
influence upon the movement of the disc 
in consequence of their inertia, when Qx a^ 



t — 




Q l^ or Qx = ^.^. 



If, therefore, a force 



P, whose arm is O A = C Bx = a, causes 
a body, whose weight is Q and whose dis- 
tance from the axis of rotation is C B = b, 
to rotate, we must reduce the latter to the 
arm a of the force P and put instead of Q, 



a 



and the mass moved by P is 

M=iP -h 



QV' 



a' 



)'9^ 



Pa' -\-Ql)' 



consequently the acceleration of the weight P is 

_ Forc e _ P Pa" 

and the angular acceleration is 
p Pa 

a 



Qy 



9- 



Pa' -^ Q¥ 

Example. — If the weight of the rotating mass is ^ = 360 pounds, its 
distance from the axis of rotation is 5 = 2,5 feet, the weight acting as 
moving force is P = 24 pounds and its arm is <» = 1,5 feet, the mass 
accelerated by P is 



M 



\P + (^ly Q\-g = 0,031 (24+^. 360\ = 0,031 . 1024 



= 31,74 pounds, 
and the acceleration of the weight is 

24 



P = 3Y^ = 0,756 feet, 



580 GENERAL PRINCIPLES OF MECHANICS. [§284 

on the contrary, that of the mass Q is 

5 5 5 . 0,756 
q = -.p = -p= = 1,26 feet, 

and the angular acceleration is 

K=^ = 0,504. 

a . ^ 

After four seconds the angular velocity is 
« = 0,504 . 4 = 2,016 feet, 
and the corresponding space described is 

^ut= Mllli_ = 4^032 feet, 

hence the angle of rotation is , 

f = ^^- . 180" = 1,2834 . 180° = 231° 1' 

and the space described by the weight P is 

vf 0,756.4^ .^.^^ 
s = ^ = ^-^r = 6,048 feet. 

§ 284- Reduction of the Moments of Inertia. — If the * 

moment of inertia of a body or of a system of bodies in reference 
to an axis passing through the centre of gravity 8 of the body is 

known, the moment of inertia in reference 
Fig. 471. ^^ ^-^^ other axis, parallel to the former, can 

easily be determined. Let S, Eig. 471, be 
the first axis of rotation, which passes through 
the centre of gravity, and D the other axis 
of rotation, for which the moment of inertia 
is to be determined ; \Qi S D — dhQ the dis- 
tance between the two axes and S Ni = x^ 
and iVi M^ = y^ the rectangular co-ordinates of an element M^^ of 
the mass of the whole body. The moment of inertia of this ele- 
ment in reference to D will be 




= M,.DM,'=^ M, [D N,' + N, M;') = M, [{d + x,Y + y'] 
and in reference to S 



and, therefore, the difference of these moments is 
= M, {d' + 2dx,-}- X,' + y,') - M, (x^' + y,') = M, d' +2M, dx^. 
For another element of the mass it is 

= M^.d'' + 2M,dx,, 
for a third it is 

= M, d"" -h 2 Ms d Xs, 

and, therefore, the moment of all the elements together is 

= (M, i- M, -{- M^ -\- ...)d'' + 2d (ifi X, + M, x,-{-MsX^ + ., .)• 



§285.] THEORY OF THE MOMENT OF INERTIA. 581 

But Ml + M2 -\- . . . is the sum M of all the masses and M^ x^ + 

J/2 ^2 + ^z ^3 is the sum M x of the statical moments ; hence it 

follows that the difference between the moment of inertia Wx of 

the whole body in reference to the axis D and its moment of inertia 

W in reference to /S' is 

Wx - pr= Md'' + %dMx, 

But since the sum of the statical moments of all the elements 
upon one side of every plane passing through the centre of gravity 
is equal to that of the moment of those on the other, the alge- 
braical sum of all the moments is — 0, and we have if 2; = 0, and 
consequently 

Wx- W=Md', 
I.E Wx = W + M d\ 

The moment of inertia of a body in reference to an eccentric axis 

is equal to the moment of inertia in reference to a parallel axis 

passing through the centre of gravity plus the ^product of the mass 

of the body by the square of the distance of the two axes from each 

other. 

We see from this that of all the moments of inertia in reference 

to a set of parallel axes that one is the least, whose axis is a line 

of gravity of the body. 

§ 285. Radius of Gyration. — It is very important to deter- 
mine the- moment of inertia for various geometrical bodies ; for the 
values thus deduced are frequently employed in the different calcu- 
lations in mechanics. If the bodies, as we will hereafter suppose, 
are homogeneous, the different portions J/i, Ji/s? etc., of the mass, are 
proportional to the corresponding portions Fi, F27 etc., of the vol- 
ume, and the measure of the moment of inertia, or as it is generally 
called, the moment of inertia, can be replaced by the sum of the 
products of the portions of the volume and the square of their 
distances from the axis of rotation. In this sense we can also 
determine the moment of inertia of lines and surfaces. If we 
imagine the entire mass of a body concentrated in one point, we 
can determine the distance of the same from the axis, if we sup- 
pose that the moment of inertia of the mass, which is thus concen- 
trated, is the same as it was, when distributed through the whole 
space. This is called the radius of gyration (Fr. rayon d'inertie, 
Ger. Drehungs- or Tragheitshalbmesser). If TTis the moment of 
inertia, M the mass and h the radius of gyration, we have 
M Ic" = W, and therefore 

"W 



^- ' M' 



582 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 286. 



We must also remember that this radius does not give a definite 
point, but only a circle, in whose circumference the mass can be 
distributed arbitrarily. 

If in the formula W\ = W ■}■ M d' we substitute W = M h"" 
and Wi = M hi", we obtain 

A^l ^^^ fC -\- d y 

I.E., the square of the radius of gyration in relation to any axis is 
equal to the square of the radius of gyration in reference to the line 
of gravity parallel to that axis plus the square of the distance of 
the two axes from each other. 



§ 286. Moment of Inertia of a Rod. — The moment of inertia 

of a rod A B, Fig. 472, which revolves about an axis X X passing 

through its middle 8, is determined in the fol- 
lowing manner. Let the cross-section of the 
rod be = i^ and half its length be ~ I, and the 
angle, which its axis makes with the axis of 
^B rotation, i.e. A 8 X,\ie ^= a. Let us divide the 
half length of the rod into n parts, the contents 

; the distances of the 




of each of which are 



n 



different portions of it from the centre S are 

-, — , — , etc., hence their distances from the 

n n n 

— I 

axis of X X, such as M N, are = - sin, a, 

n 

^— sin. a, -^ sin. a, etc., and the squares of the 



-X 



n 
latter ar 



71 



__n sm. aV II sm. ay 11 sm. a\ 



etc. 



Fl 



Multiplying these squares by the contents — of an element 

n 

and adding the products thus obtained, we obtain the moment of 
inertia of the rod 

'I sin. a^ 



T = 



Fl 

n 



~(lsin.aV ^ ^ 11 sin. aV ^ ^ R 



n 

F P sin. a^ 
n^ 



n 



■)-■■] 



(1^ + 2^ + 3^ + . . . + n% 



n' 



but since l'* + 2' + S'' + . . . + w' = y, 



387.] 



THEORY OF THE MOMENT OF INERTIA. 



683 



we have 



W = 



F f sm.' a 



Now cince Flis the volume of the half rod, which we treat as 
the mass M of the body, we have 

W 



i MV 



sm: a. 



The distance of one end of the rod from the axis X X is 

A C=BD — a — l sin. a, 

and, therefore, we have more simply 

W=lMa\ 

which formula applies to the entire rod, when we understand by 
M the mass of the whole rod. 

The moment of inertia of a mass J/j at the end A of the rod is 
Ml a" ; if, therefore, we make M^ — J M, M^ has the same moment 
of inertia as the rod. Hence, so far as the moment of inertia is 
concerned, it makes no difference whether the mass is equally 
distributed along the rod, or whether one-third of it is concentrated 
at the end A. If we put W = M Ic^, we obtain h" = \ d\ and, 
therefore, the radius of gyration of the rod is 



Iczzz a^\ =: 0,5773 . 



a. 




If the rod is at right angles to the axis 
of rotation a =■ I, and consequently 

w ^ J Mr. 

If, finally, the rod does not lie in the 
same plane as the axis of rotation, if the 
shortest distance between the axis of rota- 
tion and the axis of the rod is 

8 Si= G C, = D D, = d, 
and if the normal distances A C = B D oi 
the ends A and B of the rod from the axis 
C Z>, passing through the centre of gravity 
^S'of the rod and parallel to C^ Di is a, we 
have {according to § 284) the moment of 
inertia of the rod 

W, = W + y Md = M {d' + i a'). 



§ 287. Rectangle and Parallelopipedon — The momenta 
of inertia of plane surfaces are found in exactly the same way as 



their moments of flexure W = F^ z^ + F.^ z^ + 



We can, con- 



584 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 287.. 



sequently, employ here the values of W, found in the last section 
for various surfaces, as their moments of inertia W, 

For the rectangle A B C D, Fig. 474, the moment of inertia in 

reference to the axis X X, which runs 
parallel to one side and through the 
middle 8 of the figure, is, according to 
§ 226, _ h W 




W = 



12 



W = 



t denoting the width A B ~ G D paral- 
lel to the axis of rotation and h the 
length A D — B C of the surface. 
But the area of this surface can be re- 
garded as the mass M, and therefore 
we have 

Mil ' M Ih 
12 



~ 3 W' 



LE. equal to that of one-third of this mass concentrated at the dis- 

h 



tance S F — S G = '-^ from the axis of rotation. 

If this rectangle turns upon an axis Z Z, which is at right 
angles to its plane and which at the same time passes through the 
middle S of the figure, we have, according to § 225, 

M 
3 



^, Mh' Mlf M(¥ + ¥) 



12 



12 



12 



[©■- ©■] 



_ M (dV 

-y V2/' 



Fig. 475. 



d designating the diagonal A G = B D oi the rectangle. We can 

imagine here also one-third of the whole mass to be concentrated 

at one of the corners A, B . . . 

Since a regular parallelopipedon B E F, Fig. 475, can be decom- 
posed by parallel planes into equal 
rectangular slices, this formula is 
applicable, when the axis of rota- 
tion passes through the centres of 
two opposite surfaces. It follows 
also that the moment of inertia of 
the parallelopipedon is equal to the 
moment of inertia of one-third of 

its mass applied at one of the corners A. 



-X 




I 288.] 



THEORr OF THE MOMENT OF INERTIA. 



585 



§ 288. Prism and Cylinder. — By the aid of the formula for 
the moment of inertia of a parallelopipedon, we can also calculate 
that of a triangular prism. The diagonal plane AD F divides the 

parallelopipedon into two equal triangu- 
lar prisms, whose bases A B D, Fig. 476, 
are right-angled triangles. The moment 
of inertia for a rotation about an axis 

X X, passing through the middles C and 
K of the hypothenuses, is = jL M d^. 
Now if we employ the rule given in 
§ 284, we obtain the moment of inertia 

in reference to an axis Y Y passing through the centres of gravity 
8 and 8x 

d' 




W= j\Md' -M.GS'' 



M 



(^^-ac'^r) 



^ M 



L12 



2 V6/ i 



LB. 



W=j\Md% 

and it follows also that the moment of inertia in reference to the 
edge B His 

Wx= W ■\- M. SB' = yV Md' -h M{i dy = j\ M d' 
= \Md\ 

d denoting the hypothenuse ^ i> of the triangular base. 

For a prism A D F B, Fig. 477, whose bases are isosceles tri- 
angles, the moment of inertia in reference to an axis X X, joining 

the centres of gravity of the bases, is Wi 
= ^ M d% d denoting the side A I) = 
A F of one oi the bases; for this surface 
can be divided by the perpendicular A B 
into two right-angled triangles. Now if 
the altitude A B of the isosceles triangles, 
which form the bases, is = h, we have the 
moment of inertia of this prism in refer- 
ence to the axis Y P' passing through the 
centres of gravity of the bases 

W= '^Md' - M 1^^'^ M('^d' - i Ji') 



Fig. 477. 
D 




586 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 288. 



and, finally, the moment of inertia in reference to the edge, passing- 
through the points A and F of the bases, is 



Wi=W-\- M{^hy = M 



\6 9 "^ 9 / 



By the aid of the latter formula, we can calculate the moment 
of inertia of a regular right prism A D F K, Fig. 478, which re- 
volves about its geometrical axis. Let C A 
= C B = rhQ the radius of base or of one 
of the triangles composing the base, li the al- 
titude C' i\^ of one of these triangles A C B, 
and M the mass of the whole prism, then, ac- 
cording to the last formula, when we substi- 
tute r for d, we have 




a * "■)■ 



The regular prism becomes a cylinder, when li becomes equal 

to r, and the moment of inertia of the cylinder in reference to its 

geometrical axis is 

'r 



W= 4 M 



(f * ") 



= A Mr. 



-X 



The moment of inertia of a cylinder is equal to the moment of 
inertia of half the mass of the cylinder concentrated upon its cir- 
cumference, or equal to the moment of inertia of the whole mass at 
the distance 

h=.r^\=. 0,7071 . r. 
If the cylinder ABBE, Fig. 479, is holloiu, we must subtract 

the moment of inertia of the hollow space 
from that of the solid cylinder. Let I 
denote the length, r the radius C' ^ of 
the exterior and rg that C G of the interior 
cylinder, then we have, according to the 
above formula, for the moment of inertia 
of the hollow cylinder 
W=i (M, 7-1^ - M, r,') = 1 7T {r,' . r,'- r,' . r,') I = i n (n'- r,') I 

= l7T {r,' - r/) {r,' + r}) l = iM {r,' + ri) ; 
for the volume of the body, which may also be considered as its 

r -f- r» 
mass, is = TT (ri^ — r^) I. If r denotes the mean diameter 

and h the width r^ — r^ of the annular surface, we have 

4 




2 



W =^ M 



(^^ + 1) 



§ 289.] 



THEORY OF THE MOMENT OF INERTIA. 



587 



289. Cone and Pyramid. — With the aid of the formula 
for the moment of inertia of a cyUnder we can 
calculate those of a right cone and of a pyramid. 
Let A C B, Fig. 480, be a co7ie turning upon its 
geometrical axis and let r = D ^ == i) i? be the 
radius of its base and h = CD its altitude, which 
coincides with the axis. If by passing planes 
through it, parallel to the base and at equal dis- 
tances from each other, we divide it into n shoes, 
we obtain 7i discs, whose radii are 




r ^ r ^ r r 

~, 2 - , 3 - . . . n - 

n n n n 



h 



and whose common heiafht is - ; the Tolumes of 



these slices are 

\nl n \n I n \n I n 
and consequently their moments of inertia are 
IrV h^ /2 rV n_ l3rV 



. etc., 



h_ 

2n' 



etc. 



The sum of these values gives the moment of inertia of the entire 



cone 

2 n^ ^ 
I.E., since 1' + 2' + 3' + . 

-n r^ h 



+ 2' + 3* + . . . + n% 



n" 



+ 7^* = — - and the mass of the cone is 
5 



M 



W = 



n r* h 
10 " 



10 



7T r^ h 



r 



Mr\ 




3 10 

In like manner we have under the same cir- 
cumstances for a right pyramid ACE, Fig. 481, 
whose base is a rectangle, 

W = i Md\ 
in which formula d denotes the half I) A of the 
diagonal of the base. 

We obtain, by subtracting one moment of 
inertia from another, the moment of inertia of a 
frustum of a cone (A. B E F, Fig. 480) in refer- 
ence to its geometrical axis X X. 

If we denote the radii D A and i^by ri and r^ 
and the altitudes CD and C Ohyhi and h^^, we have 



588 



GENERAL PRINCIPLES OF MECHANICS. 

TT hi 



[§ 290. 



77 



w^=Xo^^^^^^^-^^^^^)-iOn 



(^i' - r,% 



or, since the mass is 



5 ^ f>\ 



n'), 



W = -[%M 






§ 290. Sphere. — In the same manner the moment of inertia 
of a sphere, revolving upon one of its diameters D B = 2 r, is 
determined. Let us divide the hemisphere A D B, Fig. 482, by 

planes parallel to its base A C B, into n 
equally thick slices, such as G K H, etc., 
and let us determine their moments. The 
square of the radius G K oi one of these 
slices is 

GW' = TTG' - CW.= r' - C~K% 
and, therefore, its moment of inertia is 
r 




1 TT. - (r^- CKJ 



'n r 



2n ^ ^ 

^ 2 T ^ V Tb V 

Substituting successively for C K, ~, — , — , etc., to — and 

lb ll ih III 

adding the results, we obtain the moment of inertia of the hemisphere 



I.E., 



7T T^ 4t IT T^ 



Now since the contents of a hemisphere are if = | tt r^ we can 
put W= 'i.^nr' .r' = i Mr', 

and if we consider M as the mass of the whole sphere, the formula 
still holds good. 

The radius of gyration is 

h = rV^ = 0,6324 .r; 
two-fifths of the mass of the sphere, at a distance equal to the 
radius of the sphere from the axis of rotation, has the same moment 
of inertia as the entire sphere. The formula 

W= iMr' X ... ■ 

holds good also for any spheroid whose equatorial radius is = r. 
(See § 123.) 



§ 391.] 



THEORY OF THE MOMENT OF INERTIA. 



589 



If the sphere revolves about another axis at the distance d from 
the centre, we must put the moment of inertia 

§ 291. Cylinder and Cone. — The moment of inertia of a 
circle A B D E, Fig. 483, in reference to an axis passing through 
its centre C and at right angles to the plane of the circle, since all 
points are at a distance C A — r from the axis, is 

W=Mr\ _ _ 

and consequently that in reference to a diameter X X or Y Y 

(compare § 231) is 

W,=l W= IMr'. 
On the contrary, the moment of inertia of a circular disc 
A B D E, Fig. 483, which revolves about its diameter B E, is 
found to be, like the moment of flexure of a cylinder, 



TT r 



Mr' 



~ 4 4 ' 

consequently the radius of gyration of this surface is 



^ := r V| = ' 



•J ^5 



I.E., half the radius of the circle. 
Fig. 483. 



Fig. 484. 





F 



E 



iiiiniiinni 
i iiiiin 1 
iiH iii:aimii 

i!!!llllllllll!!i|ji!;;)Sg 




G 



D 



From this we can calculate the moment of inertia of a cylinder 

ABBE, Fig. 484, which revolves around its diameter E G, which 

passes through its centre of gravity 8, Let I be the half height 

A F and r the radius C A = O B of the cylinder, then the volume 

of one half of it is — tt rW, and if we pass through it planes 

parallel to the base and at equal distances from each other, we 

7T r"^ I 
decompose this body into n equal parts, each of which is = 

ft 

I 21 

and the first of which is at a distance -, the second at a distance — , 

n n 

3 1 

the third at a distance — , etc., from the centre of gravity S. By 

means of the formula in § 284, we obtain the moments of inertia 
of these discs or slices 



590 



GENERAL PRINCIPLES OF MECHANICS. 



[§292. 



n 
n 



[ 



Ir' + 



\r' + 






= t: r"" I 



+ 2' + S'' + . 
I' 



+ n 



^)] 



.¥ 



(4 + 3) 



whose sum is the moment of inertia 
n L 4: \7iJ ^ 

of half the cylinder. This formula holds good for the ivhole cylinder, 
when M denotes its mass. 

The moment of inertia of a right prism A B D, Eig. 485, in 
reference to a transverse axis X X passing through the centre of 
gravity ^S' is determined in a similar way. Let h be the radius of 

gyration of the base ^ ^ of the prism in reference to an axis N JV, 
passing through the centre of gravity of the base and parallel 

to X X, and let I denote the half length or height C S = D S of 

the prism ; we have the required moment of inertia in reference 

to the axis X X 

W= M{k' + if). 

Fig. 486. 4 





W = ^% m(^ r^ + A ). 



In like manner we find for the right cone A B D, Fig. 486, 
whose axis of rotation passes through its centre of gravity at right 
angles to its geometrical axis C D, 

4 

§ 292. Segments. — The moment of inertia of a paraboloid of 
revolution BAD, Fig. 487, which revolves around its axis of 
revolution A C, is determined in a similar manner to that of a 
sphere. If the radius of the base is C B = C D — a, and the 
altitude C A — h, and if we divide the body into slices of the 

heisfhfc \ we have their contents 

^ n 



§ 293.] 



THEORY OF THE MOMENT OF INERTIA. 



591 



7i 1 , h 

- n ,- a , - rr 
n n n 



2 , h 3 , , 



n ' n n 

for the squares of the radii are as the altitudes or distances from 
the yertex A. From this we obtain the moments of inertia of the 
successive disc-shaped elements of the body, which are 
h 7T a^ h TT 4i a* h rr 9 a* , 

~y QXlCty 



n' %' n"^ n' %' 



n 



2 J 



n' %' n" 



and consequently the moment of inertia of the whole paraboloid is 

TT a* h n^ _ IT a* h 

^2 



W = "^/^ (1^ + 2^ + 3^ + . . . + n') 



%n' 
IT a"^ h 



6 



a' 



2 



= 1 Ma'; 



for the volume of this body is if = 



n a'^h 




This formula may be applied to a low 
segment of a sphere. 

If the altitude h of such a segment is 
not very small compared with a, we have 
for the moment of inertia of one of its. 
slices 



2 n 2 n 

n h 



hy 



- ^ .(4:r'h'-4.rh'-h h% 

2 n ^ ' 

in which r denotes the radius of the sphere. 

h 2 ^ ^ ^ 
Now if we substitute for 7^ successively the values -, — , — , etc, 

^ n n n ^ 

we obtain the mqment of inertia of the segment of the sphere 

2 nV. \n / S \ n / 4: \ n / 6 J 

= ""o^ (20 r' -16 rh -i- S h'). 

The volume or the mass of the segment of the sphere is 

M=: nh' (r-ih), 
and therefore 

generally it is sufficiently correct to put 

W=i Mh (r --f,h)--=iM (a' + I h'). 
This formula is applicable to the loh of a pendulum. 



592 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 393. 



§ 293. Parabola and Ellipse.— For the surface A B D, 
Fig. 488, of a parabola, if, instead of the surface F, we substitute 

the mass M or change F into M, and if we 
denote the chord A Bhj s and the height 
of the arc C Dhj h, we have (according to 
§ 233) the moment of inertia in reference 

to the geometrical axis X Xof this surface 

Ms' 




TFi- 



20 ' 



and that in reference to the axis Y Y, 
passing through the centre of gravity S at 

right angles to X X, is 

W, = jS% Mh\ 

Hence the moment of inertia in reference 
to an axis, passing through 8 at right angles to the surface of the 
parabola, is 

For such an axis, passing through the vertex D of the parabola, 
the moment is, since D 8 = | ^ (§ 115), 



W,= W+ M{lhy = \M 



U) 



+ -V- h^^'] 



and, on the contrary, the moment in reference to an axis passing 
through the centre C of the chord is 

This formula is also applicable to a prism whose bases are para- 
bolas, E.G. a working-beam, which consists of two such prisms 
oscillating about an axis passing through their middle C, 

The moment of inertia of an ellipse A B A B, Fig. 489, whose 

semi-axes are G A = a and C B = d,in 
reference to the axis B B, is (according 
to § 231) 

^ Tva^l Ma"" 

^. = ^r = ^-' 

and that in reference to the axis ^ ^ is 
^^^ nab' Mb' 
4 4 

hence the moment of inertia in reference 



Fia. 489. 

A 




§ 294.] 



THEORY OF THE MOMENT OF INERTIA. 



593 



to an axis, passing through the centre C at right angles to the plane 
of the figure, is 

W= W, + W, = \ M{a' + V). 

(§ 294,) Surfaces and Solids of Revolution. — The mo- 
ments of inertia of surfaces and solids of revolution can be determined 
with the aid of the Calculus by means of the following formulas. 
1) If a zone or belt P Q Qx Pi, Fig. 490, whose radius is M P 

— y and whose width is P Q — d s,\& 
caused to revolve around its geometrical 
axis A C, we have (according to § 125) its 
area 

dO = 2nyds, 
and its moment of inertia is 

y"" d — 2 7T y^ d s; 
hence the moment of inertia of the whole 
surface of revolution A P Pi in reference 
to its axis A Cis 



Fig. 490. 




W 



= 2 7T / y^ ds. 



2) For a slice P Q Qi Pi, whose volume is d V = n y^ d x, the 
moment of inertia in reference to the axis A C is (according to 

§288) 

dV.y^ _ ny* dx 

__ _ _ ^ 

and consequently the moment of inertia for the whole solid of rev- 
olution ^ P Pi is 



W 



-if'' 



d X. 



If A P is an arc of a circle, in which case the surface generated 
by its revolution is a spherical cup or zone, we have 
y"^ z=z2rx — x^ and y ds ^ r dx, 
and consequently the moment of inertia of this zone is 

W= 27r/ {^rx — x^)rdx = 2'nr \2r I xdx —J x^dxj 

= 27Tr Ir x^ —], 

or, if we substitute h for the altitude A M = x,we have 

W=27Trh'(r -|) = Mhir - |), 

since the area or mass of the zone is M = 2 n r h, 
38 . 



594 GENERAL PRINCIPLES OF MECHANICS. [§294 

For the entire surface of the sphere h — 2 r, and therefore 

W=iMr'. 

If, on the contrary, A F is, the arc of an elhpse, and conse- 
quently the solid of revolution A P P^ generated by the rotation 
of the plane surface A P M 2i segment of an ellipsoid of revolution^ 
we will have 

Uj 

and therefore its moment of inertia in reference to the axis A C '\s> 

2 a J 
— —7 / (4 cC' x' — 4:ax^ + x"") d x 



U a" x^ — a X* + —Y 



2 a' 
E.G. for the entire elliiosoid, in which case x = 2 a, 

W=j%7Tb'a=-i,t^ab'.i' = j;Md'; 

for the contents of this body are expressed hj ^ . ^n a^ = ^ n a b^ 

(compare § 123). 

3) If the belt P Q Q^ Pi revolves about an axis passing through 
A at right angles to its geometrical axis A O, we have (see § 284 
and § 291) its moment of inertia 

= d (x' + if) =2n (x' -¥ if) y d s, 
and, therefore, the moment of inertia of the whole zone J. P Pi is 



W 



= TT / (2 x" + ?/') y d s. 



4) If the entire disc P Q Q^ P^ revolves around this same axis 
passing through A, its moment of inertia is 

d V {x' + ly') =ny' {x' ^ \ f) d x, 
and, therefore, that of the entire body ^ P Pj is 

W=7T J{x^ 4- I f) f d X. 

For a paraboloid of revolution (see § 292), we have, when we 
denote its altitude A Mhj h and the radius of its base M P hj a, 

y^ _ X 
'a'~h' 
and consequently the moment of inertia in reference to the axis of 
ordinates passing through A is 

„, "^ c^ r I 1 1 d' x\ ^ ^«Vi 4 ^ OL^ ^\ 



§ 395.] 



THEORY OF THE MOMENT OF INERTIA. 



595 



or, when we substitute x = li, 

W = \-n cr h {Jr + i a') = ^ 3£ {¥ + \ a% 

since the volume of this body is = ^ tt a' h (corap. § 124). 

Hence we have the moment of inertia of this body in reference 
to an axis, passing through the centre of gravity 8 at right angles 
toAG 

W, = iM {h' + i a') - i^Y Mh'=^M{a' + i li'). 

§ 295. Accelerated Rotation of a Wheel and Axle. — 

The most frequent applications of the theory of the moment of 
inertia are to machines and instruments ; for rotary motions 
around a fixed axis are very common in them. Since throughout 
this work we shall meet with very many applications of this theory, 
we shall treat here but a few simple cases. 

If two weights P and Q act by means of two perfectly flexible 
strings upon the ivheel and axle A C D B, Fig. 491, if their arms are 

G A — a and D B — i and if the jour- 
nals are so small that the friction can 
be neglected, the machine is in equi- 
librium, when the statical moments 

P . C A, and Q . D B, are equal to 
each other, or when P a ^^ Q h. If, 
on the contrary, the moment of the 
weight P is greater than that of Q, 
or P a> Qh, P will fall and Q will 
rise ; on the contrary, if P « < § J, 
Let us therefore seek the relations of 
the motions in this case, taking, e.g.. Pay Q h. The force, 
which acts with the arm h and corresponds to the weight Q, pro- 
duces a force , whose arm is a and which acts in opposition to 

a 

the force corresponding to the weight P, so that the motive force 



Fig. 491. 



-Xr-. 




P will rise and Q will fall. 



in action at ^ is P — 



. The mass — , reduced from the arm h to 

a g 



the arm a, is ^, hence the mass moved by the force P — -^ i 

g a'' ^ c 



18 



M 



=( 



P + 



or, if the moment of inertia of the wheel and axle is FT = 



g 



and 



596 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 295. 



G h^ 
therefore the mass of the same reduced to ^ is = — ^j we have 

more accurately 



M 



= (P. 



a 



Hence the acceleration of P or of the circumference of the 
wheel is Qh 

__ motive force _ a ^ 

■P w^^^o D ^2 I r\ 7>2 , /^ 7.2 • 9 ^ 



mass 
Pa- 



P(f + Qlf + Gk'' 



Qh 



'~ Pa' + Qi'-h GJc''^^' 
hence the acceleration of the rising weight Q or of the circum- 



ference of the axle is 



q = -p 



P a~ Qh 



.gh. 



Pa' + Qh' + Gh' 
The tension of the cord, to which P is attached, is 

9 ^ 9^ 



S = P - 



P (l - ^) (see § 76), 



and that of the cord, to which Q is attached, is 



^ = «(^-f)' 



9 ^ J\ 

and, therefore, the pressure on the bearings is 



S ^ S^ = P -^ Q 



9 9 



(Pa-Q hf 

P a' + Qh' + Gk"" 

The pressure on the bearing of a wheel and axle, when in rota- 
tion, is consequently less than when it is standing still. 

Erom the accelerations p and q the other relations of the mo- 
tion can be found ; after t seconds the yelocity of P is 

V = p t 
and that of Q is 

Vx — qt\ 

Fig. 493. the space described by P is 

s = \pf 
and that by §, 

s^ = \q f. 

Example. — Let the weight upon the 
wheel, Fig. 492, be P= 60 pounds and that 
on the axle, Q = 160 pounds ; let the arm 
of the former be (7^1 = a = 20 inches 
and that of the latter I) B = 1} = Q inches, 
and let the axle be composed of a massive cylinder, weighing 10 pounds, 




§395.] THEORY OF THE MOMENT OF INERTIA. 597 

and the wheel of two rings, one weighing 40 pounds and the other 12 
pounds, and of four arms, weighing together 15 pounds ; finally, let the 
radii of the large ring A Bbe = 20 and 19 inches and those of the smaller 
one F O he = 8 and 6 inches. Required the conditions of motion of this 
machine. The motive force at the circumference of the wheel is 

P ^ = 60 — ^«o . 160 = 60 — 48 = 13 pounds, 

and the moment of inertia of the machine, when we disregard the masses 

of the ropes and journals, is equal to the moment of inertia of the axle, 

which is F52 10. 6 2 

= ^- = -^- - 180, 

plus the moment of the smaller ring, which is 

_ n,(r,^ +r,g) _ 1 3. (8^ + 6 ^) _ 

- 2 _ ^ _DUU, 

plus the moment of the larger ring, which is 

= ^2 (^s' + ^4^) _ 40 . (30^ + 19^) ^ ^^^^^ 
3 3 

plus the moment of the arms, which is, approximately, 
_ A(r^^ -7\^) _ A(r^^ + r,r^ + r ^g) _ 15 . (192 + 19 . 8 + 8^) ^^^^ ' 
- "3"(77^ rj - 3 - ^ 3 - ^^^^ ' 

hence, by addition, we obtain 

G]c^ = 180 + 600 + 15330 + 2885 = 18885, 
or, taking the foot as the unit of measure, 
18885 ,„, ,, 

= -iu = ^^^'^^- 

The whole mass, reduced to the radius of the wheel, Is 

(18885\ 
60 + 160.0,09 + ^^j. 0,031 

= (60 + 14,4 + 47,31) . 0,031 = 131,61 . 0,031 = 3,76991 pounds. 
Hence we have the acceleration of the weight P, or that of the circum- 
ference of the wheel, 

p-I q _ 13 _ 

^ P^2+ Qh2 4. Qj,2 ' ^ - 3,76991 ~ ^'^^^ ^^^* ' 

and, on the contrary, that of ^ is ^ 

g = -p = ^. 3,183 = 0,955 feet ; 
the tension on the rope to which P is hung is 

'S' = (1 --^) . P = (1 - 1^1 j . 60 = (1 - 0,099) . 60 = 54,06 pounds, 
and that of the rope supporting Q is 
S^= (1 +^^Yq = (1 + 0,955 . 0,031) . 160 = 1,03 . 160 = 164,8 pounds ; 

consequently the pressure on the bearings is S + 6\ = 54,06 + 164,8 
= 318,86 lbs., or, if we include the weight of the machine, it is = 318,86 



598 GENERAL PRINCIPLES OF MECHANICS. [§296. 

+ 77 = 295,86 pounds. At tlie end of 10 seconds P has attained the ve- 
locity 1) = pt = 3,183 . 10 = 31,83 feet, and has described the space s = 

v t ^ 

— = 31,83 . 5 = 159,2 feet, and Q has been raised up s^ = - s=0,3 . 159,2 

= 47,76 feet. 

§ 296. The weight F, which imparts to the weight Q the ac- 
celeration 

_ P ah - QW 

^ ~ P d' ^ Qlf + Glc''^' 
can be replaced by another Pi, without changing the acceleration 
of Q, when the arm of the latter is a^, in which case we have 
P,a,-Ql) Pa- Qb 

P, a,' + Q¥ + G¥~ P a' + Q I' + G k'' 

T^ ^ • ^4-1. ^-^ Pa' + Q¥+ Gh\ ,, 

If we designate the quantity p— -^ — ry-r bye, we obtain 

_ Qhjh + c) + G¥ 
0/\ c ax — — p J 

and the required arm of the lever 



(cV Ql)(b + c)G¥ 



ax = U±Vi^ P, 

We find, also, by tlie differential calculus that the greatest ac- 
celeration is imparted to Q by P, when the arm of the latter cor- 
responds to the equation PoC' — ^Qal— Qlf + G ¥, ox when 



P ' ^ \PI P 

The foregoing formulas become very complicated, when we take 

into consideration the friction of the journals and the rigidity of 

the ropes. If we denote the resistance due to both of these, reduced 

to a radius r, by F, we must substitute, instead of the motive force 

h Q h + F r 

P ft the expression P — ~ , and then we have the 

a a 

acceleration of Q 

{Pa- Fr)h-Qd 



2 



and 



^ ~ Pa' + QV -{- G¥ '^ 



_Qd-\-Fr JiQJb + FrV Qlf+G¥ 

a — p v y y p J H -p 



Example — 1) If the weights P = 30 pounds and Q = SO pounds act 
with the arms a = 2 feet and i =^ ^ ^^^* upon a wheel and axle, and if the 
moment of inertia of this machine is G^ F = 60, the acceleration of the 
rising weight Q will be 



§397.] THEORY OF THE MOMENT OF INERTIA. 599 

_ 30 . 2 . 1 - 80 . (^y _ 30-20 o o _ ?^^ 

^ ~ 80 . 2^^ + 80 . (1)^ + 60 • ^ ~ 120 + 20~+"60 * ' ~ "20" 
= 1,61 feet. 
Now if we wish to produce the same acceleration with a weight P^ = 
45 pounds, the arm of P^ must be 

/7^^ 80 . -1(1- + c) + 60 



«-^vfi)- 



but _ 200 _ 



45 

— in 
60-40 



^^^""^^ «, = 5 ± /|/ 25 - ^ = 5 ± I- . 11,358 = 5 ± 8,786 

= 8,786 or 1,214 feet. 
2) The acceleration of ^ is a maximum when the arm of the force or 
radius of the wheel is 

1 . 80 . //40\'-^ 20 + 60 4 . , /I6 , 24 4 + V40 



)Y' 20 + 60 _ 4 / 

)/ ^ 30~ ~ 3 "^ V 



= 8,4415 feet, 
and this maximum acceleration is then 

/ 30 . 1,7207 - 20 \ 31,621 

^ = (80TT3-:44157T^) ^ = 485;82 • ^ = ''^^^ ^'''' 
3) If the moment of the friction and of the rigidity of the ropes be 
i^r = 8, we must substitute, instead of ^ &, ^ 6 + -?^r = 40 + 8 = 48, 
whence it follows that 



a 



48 // 48 \^ 8 

="30"^^(3o;+3"^^'^ + "^^'^^^ "^ ^'^^^ ^^^*' 



and that the corresponding maximum acceleration is 

30 . 1,943 - 8 . 4 - 20 34,29 „^ ^ ^ ^^ ^ 

^ = 30 ■ (3,886)- r 80~ ' ^ = "SSS" * ^^'^ = ^'^^ ^""''• 

§ 297. At-wood's Machine. — The formulas for the wheel 
and axle found in § 295 are applicable to the simple fixed pulley ; 
for if we put h — a, the wheel and axle becomes 2i fixed pulley. Re- 
taining the same notations that we employed in the foregoing 
paragraphs, we have the acceleration with which P sinks and Q 
rises 

_ (P - Q) a" 

^ ^ ~ {P + Q) a" + O ¥'^' 
or, taking the friction into consideration, 

^ ~ ^ " {P + Q)d' + G¥' ^' 
In order to diminish the friction, the axle C of the pulley A B, 
Fig. 493, is placed upon the friction-wheels D E F and Z), E^ F^. 
Now if the moment of inertia of these wheels is Gx h^ and their 



600 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 297. 



radius is D B = Di B^ = «i, we have, when F designates the fric- 
tion reduced to the circumference of the axle G, 

(P - Q)a' - Far 

9'y 



p = q = 



{P + Q)a'' + G¥ -\- G, 






«i 



Fig. 493. 



for the moment of inertia of these friction rollers, reduced to their 
circumference or that of the axle of the wheel, is = 
Inversely we have the acceleration of gravity 

(P + ^) a^ + 6^ F + G, ^ 

fj __. ^1 /^ 

^ {P - 9)0"- Far 

When the difference P — Q, oi the two weights is small, the 

acceleration p is small and the motion is 
consequently very slow ; hence the resist- 
ance opposed to the weights by the air 
is unimportant, and the acceleration of 
gravity can be determined with a certain 
degree of accuracy by means of such an 
apparatus, while the determination of it by 
observations upon a body falling freely is 
impossible. Experiments of this kind were 
first made by an Englishman named At- 
wood (see Atwood's treatise on Eectilinear 
and Eotary Motion), and for this reason 
the apparatus is known as Atwood's Ma- 
chine. The scale H K, along which the 
weight P falls, serves to measure the 
distance fallen through. Erom the spaces 
fallen through and the corresponding time 
t we obtain 

% .s 
P -- 




Q 






llii^junpi 



P 



In 



f 



but if during the fall we remove the motive 
force by causing the weight L L, which is 
made in the shape of a ring and is equal to the force, to be caught 
by the fixed ring N N^, the remainder of the space s,, through which 
the weight P falls, will be described uniformly, and the velocity, 
which is determined by the time t^ (which can be observed by 
means of a good watch), is 



V = 



§ 298.] THEORY OF THE MOMENT OF INERTIA, 

and the acceleration is 



601 



V 



Si 



^ ~'i~ tt; 

If we make t^ = t = 1,^q obtain directly by the experiment 
p = Si. Substituting this value of p in the above-mentioned 
formula, we obtain the acceleration g of gravity. 

§ 298. Accelerated Motion of a System of Pulleys or 
Tackle. — The accelerations of the weights F and Q, which are 
supported by a system composed of a fixed pulley A B, and a loose 
2mUey E G, Fig. 494, are found in the following 
manner. Let the weight of the pulleys ^ ^ and 
B Ghe = G and G^, their moments of inertia G Ic^ 
and Gx Tcx, their radii A ^=^ a and D E ^= ai and 
their masses reduced to the circumference M = 



Fig. 494. 




G F 



G, h. 



— . — ^ and i/i = — . -^. If the weight P sinks 
g a^ g a^ ^ 

a certain distance 8, Q ■\- Gx rises -i s (§ 164), the 
work done is therefore P s — {Q+ Gi) ^. Now if 
in sinking the weight P has acquired the velocity v, 
then the velocity ^ is communicated to Q -\- G^, the velocity of the 
pulley A B at the circumference is v and the pulley E G acquires 

V 

at its circumference the velocity - ; for in rolling motion the mo- 

tions of translation and of rotation are equal to each other. The 
sum of the living forces, corresponding to the masses and velocities, is 



g 



Q+G, 



•(I) 



+ ■ — r . -y + 



■(I)' 



g vz/ g a' ga, 

putting the half of it equal to the work done, we obtain the equation 



P 



{Q+ G,) 



)"( 



P ^ ^ + ^i . ^>^' _, ^1 M 



V' 



2 / " V ' 4 ' a' ' 4:a;' / 2g 
Hence the velocity corresponding to the space s, described by P, is 




.,,(p-«±^) 



Q -^ G, G h 



- + 



a* 



GJcl 



602 GENERAL PRINCIPLES OF MECHANICS. [§298. 



v' 



For the acceleration p we have p s — -^, and therefore 



j,_Q+e, 



^ 4 ^ a" ^ 4 






The acceleration of ^ + 6^i is j^i = ^-, and the rotary accelera- 

tion of 6^1 is also the same. The tension on the rope B E, which 
unites the two pulleys, is 

\ ct ' 9 

for the force { P -\ — \ -^ is expended in producing the accel- 

eration of P and G\ the tension on the rope G H, which is 
fastened at one end, is, on the contrary, 

for the pulley P G is set in rotation by the difference S — Si of the 
tensions on the rope. 

Example. — The weights P — 40 pounds and Q = QQ pounds hang 
upon the system of pulleys or tackle represented in Fig. 494, and each of the 
pulleys weighs 6 pounds ; required the acceleration of each of the weights. 
The motive force is / 

P ^ — - =40 = 4 pounds. 

The masses of these pulleys, reduced to their circumferences, are 

G^^a^^G^e^3 

and the total mass is 

hence the acceleration of the sinking weight is 

4 , 16 . ^ 16 . 33,3 515,3 „ ^,, ^ , 

P = 247 • '^ ^ = W = ^47^ = W = ^'««« ^^^*' 
and that of the rising weight is 

p^ =^ = 1,043 feet 

The tension of the rope B Bis 

8= P- (p + ^) ^ = 40 - 43 . 1^ = 40 - 3,785 = 37,315 pounds. 

and that of the rope G Hia 

J3,==S-^ .{-^= 37,315 - 3 . ^ = 37,118 pounds. 



GM 



p 

E 



§ 299.] THEORY OF THE MOMENT OF INERTIA. 603 

§ 299. The motion is more complicated, when the pulley E G, 
Fig. 495, hangs only upon a cord wound around it. Let us sup- 
pose that P sinks with the acceleration^, and that Q 
Fig. 495. ^.jggg ^ith the acceleration q, then the acceleration of 
B the motion at the circumference of the loose pulley is 

Now if we put the tension of the cord A E, — 8, we 
obtain 

\ a ' g 

^ and 

for, according to § 281, we can assume, that S acts at the centre of 
gravity D of E G. Finally we have 

a ^1 1^1 ^\ 

a,' g' 
since we can assume that the centre of gravity D is fixed and that 
the pulley is put in rotation by 8. 

The last three formulas give the accelerations 

P - 8 18 -{Q -{- G,)\ . 8a,' 

a 
substituting all three in the equation q, — p — q, we obtain 

8a,' _ P- 8 8- (Q + Gr) 



ff — n i.-^ 9 n \ a. 9^ 



G,hr p ^G^'^ Q-\- o. 



a' 



whence it follows that the tension of the rope is 

^ Pa' + G¥ 



8 = 



(-^S + g^-a) (^ -^ + ^ ^y + - • 



From this value of ^S^ we find by the application of the above formula 
the accelerations of the weights P and Q, 

If we neglect the mass G of the fixed pulley and put ^ = 0, 
we obtain simply 

a ^ 2Pa' . G, Tc,' _ 2P G, k,' 

P {a,' + h,') a' + Ga' h,' ~ G, Ic,' + P (a,' + ^i^)' 
If the end of the cord A E, instead of passing over the pulley, 
is fixed, we have the acceleration ^ = 0, and therefore q, = — q^ 
and the tension 



604 GENERAL PRINCIPLES OF MECHANICS. [§299. 

for g = 0, we have 

a,' + k7 
If the rolling body Gi is a massive cylinder, we have 

Gjjcy^ _ 1 fi 

and the tension in the first case is 

2 P G, 



8 = 

8 = 



ZP + G^' 
and in the second 

G, 

r 

If in the first case the weight P must rise, we have p negative 

and 8 y Pj i.e., 

%PG,h;'yP G, Tc^ + P"" (a^ + Tc,% 
or simply 

p ^ ^^ hr 
in order that Gi shall sink it is necessary that 8 < G^, or that 

Example. — If the rope G Hoi the system of pulleys in the example of 

§ 298, Fig. 494, suddenly breaks, the rope B will be, for an instant at least, 

stretched by a force 

GJf_ 

^ w^ 2 . 40 + 3 

o = 



<^ ^'\ . i ~ (^ + tV) (40 + 3) + 1 



P = 



83.72 5976 ^„,„ 

= 25.43 + 72 = 1147 = ^'21" P"""*^'- 
Hence the acceleration of the sinking weight P is 
P-B \ /40-5,210\ ^^^ 34,79 



p + ^ 



)^=r-^if-T«^.^=^-^^-^-«'«^' 



and that of the sinking pulley is 

^ = [H^'^) ^ = (^^^") ■ -'^ = ^ • --^ = -'«^ ^-' 

and the acceleration of rotation of this pulley is 

B a.^ 5,210 „^ ^ — «^ ,, < 

gr^ = ^- . ^ = ^- . 32,2 = 55,92 feet. 



I 300.] 



THEORY OF THE MOMENT OF INERTIA. 



605 



Fig. 496. 




§ 300. Rolling Motion of a Body on a Horizontal 
Plane. — If a round body A G D, Fig. 496, is pushed forward with 

a certain initial yelocity 
c upon the horizontal 
path D E, it will, in 
consequence of the fric- 
tion upon this path, as- 
sume a motion of rota- 
tion, the velocity of 
which will gradually increase ; its acceleration jp is determined by 
the formula 

_ Force _ (p G a^ __ (j> a^ 
^ ~ M^s" ~" Mh' ~~W^' 
in which denotes the coefficient of friction, G = M g the weight, 
<j) G the friction, M Tc^ the moment of inertia and a the radius C D 
of rotation of the body. The velocity of rotation at the distance 
C D from the axis c, engendered by this acceleration in the 
time ty is 

On the contrary, the forward motion of the body suffers a re- 
tardation q, which is determined by the formula 

Kesistance (b G 

hence the velocity of this motion after t seconds is 
Vi = c — qt = c — (pgt 
Now if we put Vx = v, or 

we obtain the time after which the velocity of rotation becomes 
equal to that of translation and the rolling of the iody begins. 
This time is 

c Tc" 



t = 



(^ - 1) ^^ 



a' -hF 



c 

g 



At the end of this time the common velocity is 



a' 



<t>gt = 



a c 



and the space described by the centre C of the body is 






+ k' 



¥ 



c _ (2g'-fF)F <* 



606 GENERAL PRINCIPLES OF MECHANICS. [§ 301. 

If the coefficient of rolling friction was = 0, the hody A C 



would roll on forever with the constant velocity ^ = —^ — -j-^ upon 

the horizontal plane without coming to rest ; but since the rolling 

f C 
friction - — constantly opposes this motion (see § 192), the body, 

after describing a certain space ^i, will come to rest. At the end 

f G s 
of this space the work ^ of this friction has consumed the whole 

of the energy 

Gel G¥_ cl_ _ / a' + F \ Gc' 
% g ^ a' * 2 ^ "" V a' ) 2g 

stored by the mass of the body, and therefore we can put 

f G Si _ /a' + k'\ Gc,\ 
a ~ \ a' ) 2 g ' 
hence the space 

a^ + F Ci^ a^ c^ 



5i = 



fa '2g f{a' + ¥)2g 
is described in the time 

__2 s^ _ a? + ¥ Cx _ ac 

For a rolling ball -^ = f , and for a cylinder -^ = J (see § 290). 

In the latter case t = I - — , Cx=\ c,s = % tt-t — and 5i = I 

^ (^ g 2 (p g ^ 

a & 



f ^g 



CHAPTER II. 

THE CENTRIFUGAL FORCE OF RIGID BODIES. 

§ 301. The Normal Force. — The force of inertia manifests 
itself not only when the veloeity of a moving hody ehanges, but also 
when titer is a change in the direetion of the motion ; for a body, 



§ 301.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 



607 



by virtue of its inertia, moves uniformly and in a straight line (see 
§ 55). The action of inertia, when the direction changes continu- 
ally^ I.E. when the motion of a body takes place in a curved line, 
and particularly in a circle, will be the subject discussed in this 
chapter. 

If a material point moves in a curved line, it is at every point 
subjected to an acceleration, which causes it to deviate from its 
former direction. This acceleration has already been treated of in 
phoronomics under the name of the normal acceleration. Let the 
radius of curvature of the path of the moving body be = r and its 
velocity v, then the normal acceleration is 



Now if the mass of the point = ¥, the acceleration corres- 
ponds to a force 



P = 



r 



which we must consid^?: as the original cause of the continued 
change of the direction oi motion of the point. If the point is 
acted upon by no other (tangential) force than the normal one, its 
velocity will be constant and = c, and therefore the normal force 

r 

is dependent only upon the curvature or radius of curvature, i.E. 
smaller for a smaller curvature or for a greater radius of curvature, 
and greater for a greater curvature or for a smaller radius of curva- 
ture. When the radius of curvature is doubled, the normal force 
is but one-half as great as before. If a material point M, Fig. 497, 

is obliged to pass over a horizontal 
plane in a curved line A B D F H, 
if we neglect the friction, the point 
will have in all points the same ve- 
locity and the pressure against the 
side wall in every position will be 
equal to the normal force. While 
the point describes the arc A B 

M & 




this pressure is = 



C A 



7 > 



while 



it describes B D\i\^ — 



M & 
E~B 



M & 
for the arc i) i^ it is = — and 

GD 



608 GENERAL PRINCIPLES OF MECHANICS. [§ 302. 

M c^ 
for the qxqF H, = =, C A, £J B, G D and K F denoting the 

K F 

radii of curvature of the portions A B, B D, D F and F H of the 

path. 

§ 302. Centripetal and Centrifugal Forces. — If a material 
point or body moves in a circle, the normal force acts radially 
inwards, and for this reason it is called the centripetal force (Fr. 
force centripede, Ger. Centripetal- or Annaherungskraft), and the 
force m the opposite direction, i.e. radially outwards, with which 
the body through its inertia resists the former force, has received 
the name of the centrifugal force (Fr. force centrifuge, Ger. Centrif- 
ugal-, Flieh- or Schwungkraffc). The centripetal force is the one 
which acts upon the body inwards, and the centrifugal force is the 
resistance of the body, which acts in the opposite direction. In the 
revolution of the planets around the sun, the attraction of the sun 
is the centripetal force ; if the moving body is compelled to describe 
a circle by a guide, such as is represented in Fig. 497, the guide 
acts by its resistance as the centripetal force and opposes the centrif- 
ugal force of the body. If, finally, the revolving body is connected 
by means of a string or rod with the centre of rotation, then it is 
the elasticity of the rod, which puts itself in equilibrium with the 
centrifugal force of the body and acts as the centripetal force. 

If 6^ is the weiffht, and therefore M = — the mass of the re- 

^ . 
volving body, r the radius of the circle, in which the revolution 

takes place, and v the velocity of revolution, we have, according to 

the last paragraph, for the centrifugal force 

_, _ Mv" _ Gv' _ ^ v' G 

Jr — — — Z m X — « — • 

r g r 2 g r 

or F : G = 2 . TT- : r. 

2g ' 

I.E., the centrifugal force is to the 2veight of the tody as double the 
lieight due to the velocity is to the radius of rotation. 

If the motion is uniform, which is always the case when no 
other force (tangential force) besides the centripetal force acts 
upon the body, we can then express velocity v = c in terms of the 

duration if of a revolution by putting c = -y^ — = -^ — , and the 



§ 302.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 609 

expression for the centrifugal force becomes 



- (^JLlX — - 



A 

1 



^, , Mr = —Ti . G r. 



Since ^-n" — 39,4784, and in feet - — 0,031, we have, in a more 

\ ^ 

convenient form for calculation, the value of the centrifugal force 

^ 39,4784 ,^ , ^^^^ G r , 

P = '— . Mr = 1,2238 . -— pounds. 

The number to of revolutions per minute is often given, in which 

case, substituting for t, — , we have 

u 

SQ 47H4. 

F = '^'^ t(.'Mr = 0,010966 li'Mr^ 0,0003399 u"" Gr pounds. 
ooOO 

We have also P = 4,0243 --f = 0,001118 u' G r kilograms. 

2 7T 

Since — is the angular velocity w, we can also write 

P = oj' .Mr. 

Hence it follows that for equal times of revolution, i.e. for tJie 
same number of revolutions in a given time or for the same angular 
velocities, the centrifugal force increases as the product of the mass 
and the radius of gyration ; and if the other circumstances are the 
same, it is inversely j)Toportional to the square of the time of revolu- 
tion, or directly proportional to the square of the number of 
revolutions and to the square of the angular velocity. 

Example — 1) If a body, weighing 50 pounds, describes a circle of 3 feet 
radius 400 times in a minute, the centrifugal force is P = 0,0003399 . 
400^ . 50 . 3 = 3,399 . 16 . 50 . 3 = 339,9 . 24 = 8158 pounds. 

If this body is connected with the axis by a hemp rope, the modulus 
of ultimate strength of which is (§ 212) 7000 lbs., we should put 8158 = 

7000 . F, and therefore the cross-section of rope should hQ F = — — = 

1,165 square inches, and its diameter should be 

^_i/4^ - 

^ - y — = 0,5642 . V4,660 = 0,5642 . 2,159 = 1,22 inches. 



TV 



In order to have triple security, we must make ^ = 1,22 V3 = 
1,22 . 1,732 = 2,11 inches. 

2) From the radius of the earth r = 20| million feet, and the time of 
39 



610 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 303. 



revolution or length of day ^ = 24 hours = 24 . 60 . 60 = 86400 seconds, 

we obtain for the centrifugal force of body upon the earth at the equator 

X. . .^^ 20750000 a 2539 1 

P = 1,2238 . ^^^-^, — = -^^^, . G = K^. G, 



86400' 



864^ 
24 



290 



but if the day were 17 times as short, or -= = Ih. 24' 42", this force would 

be 17- = 289 times as great, and the centrifugal force would be nearly 
equal to the weight G of the body. At the equator, in that case, the cen- 
trifugal force would be equal to the force of gravity, and the body would 
neither fall nor rise. 

3) The centrifugal force arising from the revolution of the moon around 
the earth is counteracted by the attraction of the latter. If G is the weight 
of the moon and r is its distance from the earth, and t the time of revolu- 
tion around the latter, the centrifugal force of this body is 

G r 
= 1,2238.--^. 

Now let a be the radius of the earth, and let us assume that the force 
of gravity at different distances from its centre is inversely proportional to 
the nth. power of this distance ; we have the weight of the moon or the 
attraction of the earth (a \'* 

= '^ I? j ' 

and putting both forces equal to each other 

a \"- r 

= 1,2238 . -7^. 



(')' 



a 1 
But - = —-, r = 1251 million feet, ^ = 27 days 7 hours 42 minutes = 
r 60 

39342 minutes = 39342 . 60 = 2360520 seconds, whence 

1,2238 . 1251 1 



/ 1 Y_ 1,223^ 
\60/ ~ 393,^ 



- (6o) ' 



,4^ . 36 3600 

hence n = 2, i.e. the attraction of the earth (or gravity) is inversely pro- 
portional to the square of the distance from its centre. 

§ 303. Mechanical Effect of the Centrifugal Force. — 

If the path CAB, Fig, 498, in which the body 31 moves, is not 

at rest, hct turning upon an axis C, it 
imparts to the body a centrifugal force 
P, by virtue of which it either gives out 
or ahsorls a certain amount of mechanical 
effect. The former occurs when, in mov- 
ing in its path, it departs from, and the 
latter when it approaches the axis of rota- 
tion C. Let M be the mass of the body, 
ct) the constant angular velocity with which 
the path, e.g. a top (Fr. sabot, Ger. Krei- 
sel), turns around its axis C, and let z de- 




§ 303.] THE CENTRIFUGAL FOl-^C'J:^ OF RIGID BODIES. 611 

note the variable distance C jI of tlie body, whicli is moyiiig in 
the path CAB; we have the centrifugal force of the body 

F = G)' 31 z, 
and the work done l^y this force, while the body describes an ele- 
ment M of its path and the radius (7 if is increased by an 
amount ]\^ = ^, is 

Let us imagine the radius z to be composed of n parts, each = ^, 
then if we put z = n ^ and assume that the body begins to move 
at the centre of rotation C, we obtain the work done by the cen- 
trifugal force of the body, while the body is describing the spaCe 
C A M, during which time the distance of the body is gradually 
increasing from to z. By substituting successively in the last 
equation, instead of z, the values ^, 2 cf, 3 ^, . . . n ^, and then adding 
the values thus found, we obtain this mechanical effect 
^ = co^i¥^(^+2^+3^+...+ n^) = c^'M^'{l + 2 + 3 + ...^-7^), 
or, since 1 + 2 + 3 + . . . + ^^, when the number of members is 

great, = — , we can write 



n"^ 



A = 0)' 3f^'~ = 1 io' Mz\ 

Now the velocity of rotation of the top at the distance CM= z 
from its axis is 

V =. (jy z, 

hence we can write more simply 

' A^.lMv^'^^-G, 

2 g 

when we substitute, instead of the mass of the body, the weight 
G = Mg. 

If the body begins its motion, not at (7, but at any other point 
A without the axis of rotation, and at a distance C A ^=^ Zi from 
C, where the velocity of rotation is 

Vx = 0) Zi, 

the work ^ w^ if z^^ done by the centrifugal force while the body is 
passing from to A must be omitted, and we have the work done 
by the centrifugal force while the body passes from A to Jf 
A=1(d' Me - \ Gi'Mzx' = i(^' M{z- - z:-) 

If a body moves in a rigid path or groove, which revolves about 
a fixed axis, the vis viva of this body is increased or diminished by 



612 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 304. 



tlie product of the mass (if) and the difference of the heights due 

to the velocities of revolution ( - — and -— ) at the two ends A 

\2g 2g/ 

and M of the path. The increase takes place when the motion is 

from within outward, and the decrease when the motion is from 

without inward. 

§ 304. If a body begins its path A M B upon a top ABC, 

rig. 499, at A with a relative velocity Ci, 
and leaves the top at B with the relative 
velocity c^2, and if the velocities of rotation 
of the top in A and B are Vi and v^, the 
energy stored by the body in describing 
the path A M B, supposing no other force 
to act upon it, is 



^^ 




A = 



^ g 



G = 



Vo 



v^ 



and therefore 



c^i 



c^ = vi 



^9 



v^ 



G, 



Ca" = Ci^ + vi — vi, 

and consequently the velocity of exit is 



+ f 2 



V^ 



= ^ci + 0)^ {ri - ri), 



0) denoting the angular velocity of the top and rg and Ti the dis- 
tances A and O B of the points {A and B) of entrance and exit 
from the axis of rotation C. 

The relative velocity of exit Ci is determined in like manner,, 
when the body enters at B upon the top with the relative velocity 
^2 and moves upon it from without inwards. It is then 

c, = V^i^vi - vi) = Vci-Gi'{ri-ri). 

Since the body in describing the path A M B has, besides its 
relative velocity (c) in the path, also the velocity of rotation v of 
the path, it must be introduced at A with an absolute velocity 
A Wx =^ lOy, which is determined in intensity and direction by the 
diagonal of the parallelogram constructed with Ci and Vi, and the 
body leaves at B with an absolute velocity B tv^ = w^, determined 
by the diagonal of the parallelogram B d tu.2 v. 2, constructed with 
the relative velocities ^2 and v^. 

The energy restored, or stored, by the body in describing the 
path A M B on the top, which has been gained or lost by the 
top, is 



§304.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 613 



w^^ 



If a body should transmit all its energy -^ G to the top, while 

4> g 

describing the path A M B, the absolute velocity of exit must be 
iu<^ =. 0, and Co must be not only equal to v^^ but also exactly oppo- 
site to it ; the path must therefore be tangent to the circumference 
at i?. 

ExAiMPLE. — If the interior radius of the top, represented in Fig. 499, is 
GA = rj = 1 foot and tbe exterior one C B =: r^ = 1\ feet and if it 
revolves 100 times per minute, the angular velocity is 

TT U 10 

u=.—- = 3,1416 . -^ = 10,472 feet, 
oO o 

and consequently the velocity at tbe interior circumference is 
v^ = 0) o\ = 10,472 feet, and at the exterior one 
«j3 = w rg = 10,47 2 . 1,5 = 15,708 feet. 
Now if we cause a body, whose velocity is lo^ = 25, to enter tbe top a-t 
A, in such a direction tbat tbe angle w^ A v^ formed by its absolute mo- 
tion witb tbe direction of revolution is a = 30'', we bave for tbe relative 
velocity Cj, witb wbicb tbe body begins its motion on tbe top, 

Cj^ = -y/ + ^j2 _ 2 ^^ ^^ COS. a = 109,66 — 453,45 + 625,00 = 281,21, 

and therefore 

Cj = 16,77 feet. 
If the body is to enter witbout impact, we must bave for the angle 
v^ A c^ = (3 formed by tbe patb witb tbe inner circumference of tbe top 

sin. (3 w^ 



sm. a Cj 



m. /3 = 



or 
25 sin. 30" 



16,77 ' 
whence /3 = 48" 12' f 

For tbe relative velocity of exit Cg we bave 

Cg^ := c,-" + 'D^'' - v^' = 281,21 + 109,66 [(|)^ - 1'^] = 418,28, 
and consequently 

Cg = 20,45 feet. 
And, on tbe contrary, for tbe absolute velocity of exit W2, when tbe canal 
or groove A M B forms with tbe exterior circumference an angle 6 = 20° 
or v^ B c^ = 160", we bave 

«0g2 = Cg^ + «5g2 - 2 Cg «jg COS. S = 418,28 + 246,74 — 603,72 = 61,30, 
and consequently 

?^g = 7,80 feet. 
Finally, tbe beigbts due to tbe velocities are 

^i^ ID ■ 

^ = 0,0155 . 625 = 9,69 feet, and -^ = 0,0155 . 61,31 = 0,95 feet, 
* ^ 2 </ 

and tbe amount of mecbanical eflFect imparted to tbe top by a body, whose 

weigbt is G, while passing over tbe top, is 



614 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 305. 



(qjD '^ qjn ^\ 
-i-g ^ j a = (9,69 - 0,95) a = 8,74 G, 

or, if its weight G = 10 pounds, 

A = 8,74 . 10 = 87,4 foot-pounds. 

Remark. — The foregoing theory of the motion of a body on a top is 
directly a|)plicable to turbine wheels. 

§ 305. Centrifugal Force of Masses of Finite Dimen- 
sions. — The formulas for the centrifugal force found in the fore- 
going paragraphs are not directly applicable to an aggregate of 
masses or to a mass of finite extent ; for we do not know what 
radius r of gyration must be substituted in the calculation. To 

determine this radius, the following 
method may be adopted. Let C Z, 
Fig. 500, be the axis of rotation and 
C'Xand C T^two rectangular co-ordi- 
nate axes and let M be an element of 
the mass and M K := x, M L — y and 
M N = 2; its distances from the co-or- 
dinate planes Y Z, X Z and X Y. 
Since the centrifugal force P acts in the 
direction of the radius, we can transfer 
its point of application to its point of 
intersection with the axis of rotation. 
If we decompose this force into two components in the directions 

of the axes C X and C Y, we obtain Q = Q and R — R, for 
which we have 

Q:0 P = L: if and R : P ^ K : M, 

whence 




Q 



y-p, 



~PandiR 

r designating the distance M oi the element of the mass from 
the axis of rotation. If we proceed in the same way with all the 
elements of the mass, we obtain two systems of parallel forces, one 
m the plane X Z and the other in the plane Y Z, and each of 
which acts at right angles to the axis C Z. Employing the indices 
1, 2, 3, etc., to distinguish the yarious elements of the mass, i.e. 
putting them — M^, M^, M^, etc., and their distances, = x^, x^, x^, 
etc., we have the resultant of one system of forces 



= $1 + ft + ft + • . . = 



Xi X] Jr^i Xi^ 



n 



+ 



n 



+ 



-t 3 X3 



+ 



= G),2 ' (ifl ^1 + ifj ^2 + • • •)} 



§305.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 



615 



and that of the other 

i2 = i^, 4- i^s + . . . = w^ (if, ^1 + i/o ?/o + . . .). 

If, finally, we put the dis- 
tance C Ox, C 0<2, etc., of the 
elements of the mass from 
the plane of. X Y, = z^, ^i, 
etc., we obtain for the points 
of application U and V of 
these resultants the ordi- 
nates C U = u and C V =^ v 
by means of the formulas 

(Ci + ft -f . . .) ^^ 

and {Ri+ R<i + , . .) V = 
RiZi -\- R^z^ + . . ., whence 

Mx ^1 Zi + il/2 2:2 2^2 + • • • 




u 



and 



Vl ^1 "T" V2 ^2 "T • • • 

-/Xl Z\ ~p xLq 2^2 ~r~ • • • 



V = 



Ml yi Zi + M^ y.2 z^_ -^ . . . 



Ri + R^ + .. . Miyi + M\y, + . , . ' 

Hence we see that generally the centrifugal forces of a system 
of masses or of finite bodies can be referred to two forces, which 
cannot be combined so as to give but a single resultant when u 
and V are unequal. 

Example. — Let the masses of a system be 
M^ = 10 pounds, Jfg = 15 pounds, M^ = 18 pounds, M^ = 12 pounds, 
and their distances 

inches, ajg = 4 inches, x^ = 2 inches, x^ = Q inches, 



X, 



2/1 = 3 

z^ = 2 



u 



Vi 



Vi 



= 5 
= 3 



y, 



^4=0 



*2 — " ^3 

then the resultants of the centrifugal forces are 

Q = (j2 . (10 . + 15 . 4 + 18 . 2 + 13 . 6) = 168 . w'^ 



and 



M = 



O)' 



(10 . 3 + 15 . 1 + 18 . 5 + 12 . 3) =: 171 . 



o 



and consequently their distances from the origin G are 



u = - 



and 



10 


.0 


.2 


+ 


15 


4. 


3 


+ 


18. 


2. 


3 


+ 


12. 


6 


. 






10 


.0 


+ 


15 


.4 


+ 


18 


.2 


+ 


12.6 






10 


.3 


.2 


+ 


15 


.1. 


3 


+ 


18. 


5. 


3 


+ 


12. 


3. 






288 12 J «j . . 1 
168 =y =1.^14 inches, 



375 

ITl' 



125 

"57" ' 



2,193 inches. 



10.3 + 15.1 + 18.5 + 12.3 
The difference of these values of u and v shows that the centrifugal 
forces cannot be replaced by a single force. 



616 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 806. 



Fig. 502. 



306. If the elements of the mass lie in a plane of rotation, 

I.E. in a plane X G Y, Fig. 502, 
which is at right angles to the 
axis of rotation, as i/j, M^ . . ., do, 
their centrifugal forces will give 
a single resultant ; for their di- 
rections cut each other at one 
_x point G of the axis G Z. If we 
retain the notations of the last 
paragraph, we obtain the re- 
sulting centrifugal force in this 
case 

P=V~^'R' = wV[(-^i ^1+^2 ^2 +...)' + (if, ?/i + M,y,+ ,..y]. 

Now if G X = X and G L = y qxq the co-ordinates of the 

centre of gravity of the system of masses M = Mi + M^ 

we have 

MiXi -h M^X2 + . . . = Mx 
M^y, + J/s ^0 + . . . = My, 
whence it follows that the centrifugal force is 

P 




■\- 



Gi' 



VM' x' 



+ M'y' = 0)2 mVx' + / = G)'Mr, 



in which r = Vx^ + y'^ designates the distance G S of the centre 
of gravity from the axis of rotation G Z. 

For the angle P G X = a, formed by this force with the axis 



C X, we have 



R My y 
tana, a = -^ = ^r^ = ^; 
^ Q Mx x' 



Fia. 503. 
z 



consequently, the direction of the centrifugal force 2^asses through 
the centre of gravity of the system, and that force is precisely the 
same as it would he if all the elements of the mass ivere concentrated 
at the centre of gravity. 

For a disc A B at right angles to the axis of rotation Z Z, 

Fig. 503, the centrifugal force is also = 
0)"^ M r, if M denotes its mass and r the dis- 
tance G S of its centre of gravity from the 
axis. If the centres of gravity of the ele- 
ments of the mass of a body lie in a plane of 
rotation, or if this plane is a plane of symme- 
try of the body A B F F„ Fig. 504, the cen- 
trifugal forces of the elements of the mass of 
the body can be combined so as to give a 
single resultant acting at the centre of gravity of the body, and 




-z 



§ 306.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 



617 



this resultant corresponds to the distance of this point S from the 
axis of rotation and can therefore be determined by the formula 

Fig. 504. Fig. 505. 

Z 





In order to find the centrifugal force of a body ABBE, 
Fig. 505, let us divide it into disc-shaped elements by planes per- 
pendicular to the axis Z Z, and then find their centres of gravity 
8x, Si, etc. ; we can then determine by the aid of the latter the cen- 
trifugal forces, by decomposing these into their components in the 
directions of the axes C X and C Y and by combining the compo- 
nents m the plane Z C X,^q obtain the resultant ft and by com- 
bining those in the plane Z C Y,^q obtam their resultant R. 

If the centre of gravity of all the discs lie in a line parallel to 
the axis of rotation, we have x ^ x-^ = Xs, etc., and ^ = ^/i = y^i, etc., 
and therefore r = Tx — r^, etc., whence it follows that the centrif- 
ugal force of the whole body is 

P — w' (ifi r + i/2 r +...) = w' Mr, 
and that the distance of the point of application from the plane 

XFis 

_ (i/, Zx + M.2Z. + . . .) r _ M^Zy + M<iZ^ + . . . 

{M, + M^ -\- . , .) r ~ J/i + if 2 + . . . 

From these equations we see that the centrifugal force of a body, 
which can be divided into discs, whose centres of gravity lie in a 
line parallel to the axis of rotation, is equal to the centrifugal force 
of the mass of the body concentrated at its centre of gravity, and 
the point of application of this force is at the centre of gravity. 

Hence we can find in this manner the centrifugal forces of all 
symmetrical bodies (see § 106), whose axis of symmetry is parallel 
to their axis of rotation, and also that of all solids of revolution, 
whose geometrical axis is parallel to the axis of rotation. If the 
axis of rotation and the geometrical axis coincide the resulting 
centrifugal force is = 0. 



z = 



= z. 



618 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 307. 



Example. — The dimensions, heaviness and strength of a mill-stone 
A B D E, Fig. 506, are given ; required the angular velocity w when the 
stone is torn apart by the centrifugal force. Putting the radius of the 

millstone = r^, the radius of its eye 
= rg, its height A B = H L = I, 
its heaviness = y and the modu- 
lus of ultimate strength = K, we 
have the force necessary to tear the 
stone apart in a diametral plane 

P=2 (r, - r,) I K, 
the weight of the stone 

and the radius of rotation for each 
half of the stone, i.e. the distance 
of its centre of gra\dty from the 




— P 



— Z 



axis of revolution (see § 114), 



r = 



4 



At the moment of tearing apart the centrifugal force of one-half the stone 
is equal to the breaking load of the stone, and we have 

Gr 



0) 



1 



ff 



I.E., 



6;=.f(r,= 



I y 



r,) I K. 
Cancelling 2 Z on both sides of the equation, v^e have 



^cjK 



Now if r^ = 2 feet = 24 inches, rg = 4 inches, K = 750 pounds and 

the specific gravity of the stone = 2,5, or the weight of a cubic inch of it 

62 4 2 5 
y = — ' ' ' = 0,09028 pounds, we have the angular velocity, when 

1728 

the tearing begins, 



= / 



3 . 12 . 32,2 . 750 

688 . 0^09028 



./: 



3375.16,1 



43 . 0,09028 



= 118,3 inches. 



If the number of revolutions in a minute = u, we have cj = 



2 IT u 
~60~ 



and 



30 CO 
inversely u = — ^j-, or m this case, = 



30 . 118,3 



TT 



11291^. 



Generally the number of revolutions of such a stone is 120 or about nine 
times less. For a fly-wheel we can put r^^ + ^i ^2 + ^2^ = S r"^, r denoting 
the radius of the middle of the ring, and consequently we have 






r" y 



§ 307. If all the parts M^, M^ of a system of masses, Fig. 507, 
or the centres of gravity of the elements of a body are in a plane 



307.] 



THE CENTRIFUGAL FORCE OF RIGID BODIES. 



619 



Fig. 507. 




passing through the axis of rotation, the centrifugal forces form a 
system of parallel forces and can be referred to a single force. Let 

the distances of the elements of the 
mass from the axis of rotation Z ZhQ 

0, M, = r,, 0, M, = r„ etc., 
then the centrifugal forces are 
Pi = 0)2 M, r„ Pa = w' M, r,, etc., 
and their resultant is 
Pi = G)' {M, r, + M,r, + .. ) 
= ui' Mr, 
r denoting the distance of the centre 
of gravity of the whole mass M from 
the axis of rotation. The distance 
of the centre of gravity from the axis 
of rotation must be considered here as the radius of rotation. In 
order to find the point of application of the resulting centrifugal 
force P, we substitute the distance of the elements of the mass 
from the normal plane, viz., C Oi — Zi, C Oct = z^, etc., in the formula 

^ ^ _ _ M, r,Zi + M^r < ,z,-\-... 

M,r, + M,r, + ... 

By the aid of the formula P = co"^ Mr the centrifugal forces 
of solids of revolution and of other geometrical bodies can be deter- 
mined, when the axis of these bodies is in the same plane as the axis 
of revolution. 

For a 7'od A G, Fig. 508, whose length is A C = I and whose 

angle of inclination A C Z to the axis 
of rotation is = a, we have 

r = K S = I I sin a, 
and consequently the centrifugal force 

P = G)"" . h Ml sm. a ; 
but in order to find the point of appli- 
cation of this force, we must substi- 
tute in the expression 
, M 



Fig. 508. 




0) 



n 



X sm. a . X COS. a 



= <y^ 



M , . 

— X sm. a COS. a 
n 



M 



for the moment — of the rod successively, instead of x, the ele- 



620 GENERAL PRINCIPLES OF MECHANICS. [§307. 

ments -, — , — , etc., and add the expressions thus obtained to- 
n n n ^ 

gether. In this manner we find 

M f 

P u = i^"" — sin. a cos. a — „ (1' + 2' + 3' + . . . + n") 

= \ (x!^ Mf sin. a cos. «, ^^ 

hence the arm C L = 0^ or 

u = I G)"^ M V" sin. a cos. a : ^ (joi^ M I sin. a = | Z cos. a, 

and the distance of the point from the end C of the rod, which 

lies on the axis, is 

C = ?^l. 

If the rod A B, Fig. 509, does not reach the axis, we have 

P = \d^ Fl^ sin. a — J- w'' F 4' sin. a 
= ^(li^ F sin. a (l^^ — //), 
and the moment 

P u = ^ oy^ F sin. a cos. a (Zj^ — ^2^) ; 
for the mass of C A (= cross-section multiphed by the length) is 
= Fli and the mass oi C B, = F l^. 

It ' follows, therefore, that the distance of the point of applica- 
tion from the point of intersection C with the axis is 

= 1 ^^^— ^ otCO = 1 + ^^\Z !'^\ 



I denoting the distance C S of the centre of gravity and li — l^ the 
length of the rod. 

Z Fig. 509. Fig. 510. 





This formula holds good also for a rectangular plate A B D E, 
Fig. 510, which is divided into two similar rectangles by the axial 
plane C Z, and whose plane is at right angles to this axial plane ; 
for the points of application of the centrifugal forces of the slices, 



308.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 



621 



obtained by passing planes through it perpendicular to C Z, are 
in the medial line F G. Now if the distances C F and C G of the 
two bases A B and F E from the origin C are k and k, we have 
here also 
\ '^ hhy n n 2 ^1^ — ^2^ __ 7 I (^1 ~ ^2) 



I + 



Fig. 511. 



l,'-k' ' ' 12 1 
In like manner the centrifugal force of a right cone A B D, 
with a circular base, Fig. 511, which turns about an axis C D 

passing through its apex, is found by 
substituting in the formula P = o)^ M r 
for r the distance K 8 of the centre of 
gravity S of this body from C Z. If h 
denote the altitude KI) of the cone, and 
a the angle B C Z formed by the base 
of the cone with the axis of rotation, 
we will have 

K 8 = D~8 COS. D 8 K-= IJi cos. a, 
and consequently the required centrifu- 
gal force is 

P = w'iff hcos.a. 

The point of application of this 

force is determined by the co-ordinates 

D L = tc and L = v, for which we 

find with the aid of the Calculus, under 

the supposition that the axis of rotation C Z does not pass through 

the cone, the following expression 

u = i h sin. a ] 




i-h^. 



and 






r denoting the radius K A = KB of the base. 

§ 308. If all the different parts of the body lie neither in a 
plane normal to the axis of revolution, nor in one containing that 
axis, the resulting centrifugal forces 

Q = 6)2 (Ml aji + i/2 ^2 + . . .) and i? — w^ (M^ y^ + M.y^ + . . .) 
wiU not give a single force, but it is possible to replace them by a 
force 

P=zVQ' + R' = (^'Mr, 
applied at the centre of gravity, and by a couple composed of Q 
and R. If we apply at the centre of gravity four forces + Q and — Q 
as well as + i? and — R, which balance each other, the positive 
forces will give the resultant 



622 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 308. 



Fig. 513. 



while the negative ones — Q and — B, together with the centrifu- 
gal forces applied at U and V (see Fig. 501) form the couples 
(ft — Q) and (E, — B), which can be combined so as to form 
a single couple. 

In order to understand better this referring of the centrifugal 

forces of a revolving body to 
one force and one couple, 
let us consider the following- 
simple case. The rod A B, 
Fig. 512, which revolves 

about the axis Z Z, is paral- 
lel to the plane Y Z and its 
end A reposes upon the axis 
C X, Let us denote the 
length A B of the rod by I 
its weight by G, the angle 
A B B^, formed by the rod 
with the axis of rotation, by 
a and its distance O A from 
the plane 1^^, which is also 
its shortest distance from 

the axis Z Zhj a. Now if 

¥ 

E IS an element — of the rod, 
n 

and y == A E^, the horizontal projection of its distance A E from 

the end A, we have the components of the centrifugal force Pi of 

this element 




M 



M 



M 



M 



Oi = 0)2 . — . C A = 6)2 . — a and i?i = o)^ . — .AEi = ui'. — v. 

^ n n n n ^ 

and their moments in reference to the plane X G Y of the base, 
since the distance of the element from this plane X 1^ is 

ExE — A Ex cotg. cl = y cotg. a, are 

M M 

Oi 2;i = w^ . — , G A ,ExE = d^ . — ay cotq. a and 

M 

B^ 0j = 0)2 . — y'^ , cotg. a. 

The resultant of all the components parallel to XZis 

M 

Q — Qi + O2 + . . . = w . 0)2 . — a = o)\ M a, 

n 



§ 308.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 623 

and its moment is 

M 
Qu = Qi Zi + §2 ^2 + • . . = w^ . — a cotg. a (yi + Vs + . . .)> 

lb 

I sin. a 2 1 sin. a 3 1 sin. a . 

or, smce yi = , y^ = ,^3 = — ,etc.,aiidcor^.a. 

it n n 

sin. a — COS. a, we have 

Ml' M I 71^ 

Qu =^ (m^^ , — a COS. a. . - (1 + 2 + 3 -H . . . + w) = w^ — a cos. o.- . — 

n n n n 2 

= ^ G)"^ . M a I COS. a. 

The distance of the point of application of this component from 

the plane X ^T of the base is 

^ ^ h (i)'^ M al COS. a , , 

iSi iS = u = T-^/ = ^ i COS. a, 

I.E., this point coincides with tJie centre of gravity of the rod. 
The resultant of the components parallel to Y Zis 

M 

R =z Ei + R, + . .. = 0)' . — {yi -\- y, + .. .) 

n 

2 M I sin. a ^^ , 2 71^7 • 1 -4. i.- 

= G)\ — . --— = -S (^ Ml sm. a, and its moment is 

n n 2 ^ 

M 

72 -y = 0)^ — . cotg. a (?/i^ -\- y^ + ' ' ') 

71/ 

, M ^ ((Isin.af (2 I sin. aY \ 

z= u)' ,— . cotg. a{^ -^ + -^^ , — L + . . .) 

n ^ \ n w } 

M V 

= 0)^ . — . -^ isin. aV cotq. a (1 + 4 + 9 + . . . + ^^) 
n n 

, M f . n' 

= o)\ — . —^ sm. a COS. a . —~ 
n n" 3 

= \ 0)'^ M r sin. a cos. a. 
Hence the distance of the point of application ot this force 
from the plane X 1^ is 

^ ^ 1 w^ Mr sin. a cos. a ^ , 

Oi O = V = - — - — rn^i — = = 1 ^ COS. a, 

^ (m)' Ml sin. a ^ ' 

I.E. this point lies at a distance (| — ^) I cos. a = ^ I cos. a verti- 
cally above the centre of gravity, or, in general, S = ^ of the 
length of the rod A B. 

From the two components Q = o)' M a and R = r, uy" Ml 
sin. a, it follows that the final resultant, which acts at the centre 
of gravity of the rod, is 

P = V Q" -^ R^ = Gi' MVa' -^ 1 rsin.a% 
that the couple is {R, — R), and that its moment is 



624 GENERAL PRINCIPLES OF MECHANICS. [§309,310. 



Ji,SO = rfO)'^Ml sin. a , ^ I cos. a 

= j^2 ^^' ^ ^' ^**^- ^ ^^^- ^ = ih ^^ ^ ^' ^^^' ^ 0^' 



§ 309. Free Axes. — The centrifugal forces of a body revolv- 
ing uniformly upon its axis generally exert a pressure upon the 
axis, yet it is possible for these forces to balance each other, in 
which case the axis is subjected to no pressure from them. As ex- 
amples of this case we may mention solids of revolution turning 
around their axis of symmetry, or geometrical axis, the wheel and 
axle, water wheels, etc. If a body in this condition is acted upon 
by no other forces, it will remain forever in revolution, although 
the axis is not fixed. This axis of rotation is therefore called a 
free axis (Fr. axe libre, Grer. freie Axe). From what precedes, we 
know the conditions, which are necessary when an axis of rotation 
becomes a free axis. It is necessary that not only the tAvo re- 
sultants Q and R of the forces parallel to the co-ordinate planes 
X Z and Y Z, but also that the sums of the moments of each of 
the two systems of forces shall be — 0, whence it follows that 

1) ifi a^i + J/s ^2 + • . • = 0, 

2) ifi yi + ifs ^2 + . . . = 0, 

3) Ml Xi Zi + Jfg ^2 ^2 + . . . = and 

4) M,yiZi + M^y^Zi + ... = 0. 

The first two conditions require the free axis to pass through 
the centre of gravity of the body or system of masses. The two 
latter, however, give the elements required for determining the po- 
sition of this axis. It can also be proved that every body or system 
of masses has at least three free axes, and that these axes are at 
right angles to each other and intersect each other at the centre of 
gravity of the system. 

The higher mechanics distinguishes from the free axes other 
axes, which may intersect each other at any point of the system and 
which are called principal axes (Fr. axes principaux, Ger. Haupt- 
axen). It is also proved that the moment of inertia of a body in 
reference to one of the principal axes is a maximum, and in rela- 
tion to the second it is a minimum, and in relation to the third it 
has a mean value, and that for a point which lies in the free axes 
the principal axes are parallel to the free axes, i.E. to the principal 
axes passing through the centre of gravity. 

§ 310, Free Axes of a System of Masses in a Plane. — 

If the parts of a mass are in a plane, e.g., if they form a thin plate 



§ 310.] 



THE CENTRIFUGAL FORCE OF RIGID BODIES. 



625 




or plane figure, then the straight line, passing through the centre 
of gravity of the entire mass at right angle to that plane, is 
a, free axis of the mass ; for in this case the mass has no radius 
of rotation, and therefore the only possible centrifugal force is = 0. 
In order to find the other two free axes, we employ the following 
method. Let S, Fig. 513, be the centre of gravity of a mass and 

let U U and F F be two co-ordinate 
axes in the plane of the mass and let us , 
determine the elements of the mass by 
means of co-ordinates parallel to these 
axes, E.G. the element M^ by the co-or- 
dinates M^N=Ux and M^ = Vi. JSTow 

if X X is one free axis and Y Y an axis 
at right-angles to the same and if the 
angle USX, which the free axis makes 
with the axis of co-ordinates S U and 
which is to be determined, = (f), then 
putting for the co-ordinates of the elements of the mass in refer- 
ence to Jl X and Y Y, Xi, 0^2 .. . and yi, ^2? • . •? e.g. for those of 
the mass M^ 

Ml X = Xi and M^ L = y^, 
we obtain 

Xi=Mx K=S R + RL=S Ocos.(fi+ MiSin,<p=UxCOS.(f) + ViSiH,(l>, 
y, = M,L= -OR +0F^ - SOsin.<l)+ OM,cos.<p 

= — Ui sin. (j> -h Vi COS. <p, 
and therefore the product 

Xi 3/1 = {uy COS. </) + ^1 sin. (p) . {— Ui sin. (p + Vi cos. 0) 

= — (tCi^ — Vi) sin. 4> COS. + Ui Vi (cos. 0^ — sin. 0'), 
or, since sin. cos. = 2 sin. 2 and cos. (jy' — sin. 0^ = cos. 2 0, 

^\y\~— i (wi' — '^^i'^) sin. 2 + Wi Vicos. 2 0, 
and therefore the moment of the element Jfj is 

M 

-^1 ^1 2^1 = ^ {ui — Vx^) sin. 2 (f) -{- Mx Ui Vi cos. 2 0, ) 

and in like manner the moment of the element i/2 is 

M^ 
J/2 x^yc^ = (u^^ — v^) sin. 2 (f) -h Mc^ u^ Vc^ cos. 2 0, etc., 

and the sum of the moments of all the elements or the moment of 
the entire mass itself is 

M^x^yx + J/2 ^'2 ^2 + ...= — i sin. 2 [(ifi Ux' + M^ n^ + . , .) 
— {Mx Vx ■\- M^v^ -\- ., .)] + COS. 2 (Mx Ui Vx + M^u^v^ + ,. .)*^ 
40 



626 



GENERAL PRINCIPLES OF MECHANICS. 



[§311. 



In order that X X shall be a free axis, this moment mnst be 
= ; we must therefore put 
\ sin. 2 (f) [(ifi th' + M, u,^ + ...)- {M, v,^ + M,v,^ + . . .)] 
— COS. 2 (3/i ti^ Vi + iltfg Ui V.2 + . ..) = 0, 
from this we obtain the equation of condition 

_ sin. 2 (f) _ 2 {M^ Ui Vi + M^ Ui v^ ■{- . . .) 

tang. 2^- -^—^ - (ji/^^^;^ ^M,u,' + ,..)- {M, v^' + M, v7+TT7) 

_ Double the moment of the centrifugal force 
Difference of the moments of inertia. 
This formula gives two values for 2 0, which differ from each 
other 180°, or two values of differing 90° from each other ; this 

angle therefore determines not only the free axis X X, but also 
the free axis Y Z" perpendicular to it. 

§ 311. The free axes of many surfaces and bodies can be given 
without any calculation. In a symmetrical figure, E.G., the axis of 
symmetry is a free atxis, the perpendicular at the centre of gravity 
is the second, and the axis at right-angles to the surface of the 
figure the third free axis. For a solid of revolution A B, Fig. 514, 
the axis of rotation ^ Z is one free axis and in like manner every 
normal X X, YY. . .to this line and passing through the centre 
ot gravity is another. For a sijhere every diameter is a free axis, and 
for a right parallelopipedon A B D, Fig. 515, bounded by 6 rectan- 

FiG. 514. Fig. 515. 

z ^ 



'X' 





gles they are the three axes XX, Y Y and Z Z, passing through 
the centre of gravity perpendicular to the sides B D,A B and A D, 
and parallel to the edges. 

Let us now determine the three axes for a rhomhoid A B C D, 
Fig. 516. We begin by passing two rectangular co-ordinate axes 
U CTand V T through the centre of gravity, so that one is paral- 



§ 311.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 627 

lei to the side A B of the rhomboid, and by decomposing the rhom- 
boid by parallel lines in 2 ^ equal strips, such as F G. Now if one 
side A B = 2 a and the other A D = 21? and the acute angle ADC 
between two sides = a, we have the length of the strip B G, 

situated at a distance S B = x 
Fig. 516. ^^^^^ u^ 

^= K G + E K = a + X cotg. a, 
B and that of the other part B F 



Y 



A 




Yfif,m.r;,r, 'f^ ^ ^^mmp^ -^ ^ =: a — X cotq. a, 

/=^^^^ U -^ . 

and since - sin. a is the width of 

n 

both, we have the areas of these 

strips 

h sin. a . , \ -, b sin. a , , . 

= ia-rx cotq. a) and {a — x cotq. a) ; 

and consequently the measures of the centrifugal forces of the two 
portions in reference to the axis V V are 

i sin. cb , , , N 1 / , , \ ^ 8in. a , , x, 

= \a-^x cotg. a) .\{a -V X cotg. a) = — (a -{• x cotg, ay 

and 

h sin. a ^ \2 

—^ (a ~ X cotq. ay, 

2n ^ ^ / ^ 

and their moments in reference to the axis U U are 

h sin. a ^ \2 J i si?i. a , ^ x, 

--r [a -\- X cotq. ay x and -— (a — x cotq. aV x. 

2 n ^ 2 01 ^ ^ ' 

Since the two forces act in opposition to each other in reference 
to y F, by combining their moments we obtain the difference 

I X sin. a , n2 / ^ x2t ^ z . 

— \\a + X cotg. a) — (a — x cotg. ay] = - ad x cos. a. 

Tj? T, iv X • xi • £• 1 • 1 ^ sin. a 2 h sin. a 
It we substitute m this lormula successively , -, 

n n 

3 h sin. a ^ -,-, ,-, -,. -, . , 

, etc., and add the results, we obtain the measure of the 

moment of the centrifugal force of one-half the parallelogram 

2ab ¥sin.^a^^^ ^„ ^„ „ n^ 

cos. a. , — {r + 2'-\-S'+... + n') = 2ab'sin.'acos.a. 



= I « Z>^ si7i.- a cos. a, 
and for the Avhole parallelogram we have 



628 GENERAL PRINCIPLES OF MECHANICS. [§ 31L 

Ml Ui Vi -{- M<i Uci V2 -\- . . . = i a ¥ sin^ a cos. a. 
The moment of inertia of one strip i^^ 6^ in reference to V F^is 

_ i sin. a I {a + x cotg. a)^ ^ (a — x cotg. a)'' 
\ "3 ^ 



fi 



+ 



2 h sin. a 2 ^ 2 \ ^ di . , . 09 ^ox 

= — ^^ {a + 3 a x^ cotg. a ) = ^ — sin. a (a^ + 3 x^ cotq? a). 

6 n n if J 

ci 1 •', ,' n • 1 ^ sin. a 2 b sin. a S b sin. a , 

Substituting lor x successively — , , , etc., 

n n^ n 

and summing the resulting values, we obtain the moment of inertia 
of one-half the rhomboid, which is 

= ^ a b sin. a (a^ + b"^ cos.^ a), 
and for the whole rhomboid it is 

= ^ ab sin. a (a^ + b'^ cos.'' a). 

In reference to the axis of rotation TJ U the moment of inertia 
of the parallelogram is 

b sm cb 
=1 ^ ab sm. a — -—^ — =z ^ a¥ sin.^ a (see § 287), 

o 

and the required difference of the moments is given by the equation 

(ifi Ui' -h M,U,' + ...) - (Jfi Vr' + M,v,' + ,, .) 

= ^ ab sin. a {(f + b'^ cos.^ a) — ^ a¥ sin.^ a 
= ^ ab sin. a [a^ + b'^ {cos.^ a — sin.^ a)] 
= ^ ab sin. a (a^ + b"" cos. 2 a). 

Finally, we have for the angle U S X = </>, which the free axis 

X X makes with the co-ordinate axis U U ov with the side A B, 
according to § 310, 

tang. ^ cp - ^^^ ^^^ ^ m,u,' + ...)- {M, v,' + M,v,' + . . . ) 
2 . ^ aF sin.^ a cos. a _ b^ sin. 2 a 



^ a b sin. a (a^ + b^ cos. 2 a) a^ + W cos. 2 a 

Por the rhombus a — b, and 

sin. 2 a 2 sin. a cos. a 2sin.acos.a 

tang.2(b= 7- x— = z ^ ^^- = —^ = tang, a, 

l + cos.2a 1 -{- cos. a — sm.^ a 2 cos. a ^ 

ex T a 

or 2 (p = a and cf) = ~. 

Since this angle gives the direction of the diagonal, it follows; 
that the diagonals are free axes of the rhombus. 



§ 312.J THE CENTRIFUGAL FORCE OF RIGID BODIES. 



629 



Example. — The sides of the rhomboid A B G D, Fig. 516, are J. ^ = 
2 « = 16 inches, aud i? (7 = 2 5 = 10 inches, and the angle A B C— a = 
60° ; what are the directions of the free axes ? 

Here we have 



tang. 2 <l> = 



5^ sin. 120'' 



25 sin. QO" 



25 . 0,86603 



8' + 5"' COS. 120° 64 — 25 cos. 60° 64-35 . 0,5 
= 0,42040 = tang. 22° 48' or tang. 202° 48' ; 

hence it follows that the angles of inclination of the first two free axes to 
the side A B are = 11° 24' and 101° 24'. The third free axis is perpen- 
dicular to the plane of the parallelogram. These angles determine the free 
axes of a right parallelopipedon with a rhomboidal base. 

§ 312. Action upon the Axis of Rotation.— If a material 
point M, Fig. 517, revolves with, a variable motion around a fixed 

axis C, the latter must coun- 
^^^' ^^'^' teract not only the centrifu- 

gal force, but also the force 
of inertia of this point. While 
the centrifugal force acts ra- 
dially outwards, the force of 
inertia acts tangentially either 
in the opposite or in the same 
direction as the movement of 
rotation, according as the ac- 
celeration of this motion is 
positive or negative (Ketard- 
ation). We can therefore as- 
sume that the centrifugal force 



— X 




M N = C N = N acts directly upon the axis C, and that the force 
of inertia if P = — P is composed of a couple {P, — P) and an 
axial force, — P, and consequently the entire force, acting upon the 

axis, C R = R is represented by the diagonal of a right-angled 
parallelogram formed of N and — P. If r is the distance C M oi 
the mass M from the axis of rotation C, w the angular velocity and 
K the angular acceleration, we have, according to § 302 and § 282, 

N=oi'Mr 
and P = K M r, 

and therefore the required resultant is 

R = ViV^ + P' = V(o' + k' . Mr, 



630 GENERAL PRINCIPLES OF MECHANICS. [§312. 

and for the angle U C J^ = (p, made by this force with the 
direction M of the centrifugal force, we have 

i w. -P P ic 

tang,cl> = -^- = - -^ =. - — ,. 

Since in consequence of the acceleration k, cj is variable, the 
centrifugal force JV and the resultant E are variable. 

In order to combine the centrifugal forces and the forces of 
inertia of the masses M^, M^, etc., we decompose each of these forces 

into two components parallel to the directions of two axes X X and 

Y Y, then if we combine them by algebraical addition, so as to 
obtain two forces acting in the direction of each axis, we have only 
to determine the resultant of these two forces. If x and y are the 
eo-ordinates C K and C L of the material point M in reference to 

the co-ordinate axes XX and Y Y, we have the two components 
of the centrifugal force X 

r 

and, on the contrary, those of the force of inertia 

P,=.y P — K. Jf ^and 

P, = ~P = kMx, 
r 

and therefore the entire force in the axis X X is 

Q = N, + P, = 0)" Mx + K My, 

and that in the axis F F is 

R :=. N, - P^ = i^'' M y - K M X, 

If we have a system of points or masses M^, if^, etc., which are 
revolving about a fixed axis C, Fig. 518, and if the co-ordinates of 
these points in reference to the axis X X are 

0-0-1= Xiy C x5l2 ^^=^ ^2? etc., 

and those in reference to the axis Y Y are 

C Li = ^1, C L^ — ?/o, etc., 
the entire force in the direction of the first axis is 

Q — Gi'M^x^ + K M^y^ -\- G)"" M^ X.2 + K M, y^ + . . ., i.e. 

g = 0)' {M^ Xy + M^x^ + ...) -\- K {M^ ^1 + ^2 ^2 + . . '), 
and that in the direction of the other axis is 

B z=z G)^ [M, y, -{- M, y, -\- . . .) — K {M, x, + M<,x<i + ., .). 



%S12.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 



C31 



Now if we denote the entire mass ifi + i¥o + . . . by M and 
the co-ordinates of its centre of gravity in reference to the axes 
JT X and Y Yhj x and y, we have (see § 305) 

Fm. 518. 
— T 



— X 




M^Xi + M^Xi + ...= Mx 
M^y^^- Mo.y^ + ... = My, 
and therefore, more simply, 

Q = oy" Mx + K My and 

E = o)'My — KMx, 
From Q and B we obtain the resultant 

S = V Q' + R% 
and for the angle X C 8 = of its direction 

tang, (p = -j- 

Since M x and My are the statical moments of the centre of 
gravity, it follows that in determining the pressure 8 upon the axis 
of a system of masses, situated in one and the same plane of revo- 
lution, we can consider the whole mass to be concentrated at the 
centre of gravity ; and since the distance of the centre of gravity 
of the system of masses from the axis of rotation is 

r = Vx" -r y\ 
we have also 

8 = 4/[(w' Mx + K MyY + (w' My - k Mx)*] 
= M^/[Gi' {x' + f) + k' {x' + y')] 

= MVgj^ 



kWI 



= Vc 



Mr 



x/ -i- y = roj^ + fc' 

Remark. — If a triangle ABC, Fig. 519, revolves about its corner (7, 
and if the other comers A and B are determined by the co-ordinates 



632 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 31^. 



(ajj, y^) and (ajg, y^), we have, according to § 113, the co-ordinates of its 
centre of gravity 8 



Fm. 519. 




and 



C8, ==y 



X. + (B 



2 



3 



and the mass, if we measure it by its super- 
ficial area, is 



M^ 



^1 Vz ^8 Vx 

2 



Its moment of inertia in reference to the axis 
of rotation C can be determined by the for- 
mula 



W 



6 \x, —x^ 



+ 



yx-y% 

Vx - Vz 



M 



= -g- {^x + a?! «2 + ^z + Vx + VxVz + Vz")' 

This formula is also applicable to a HgJit prism, whose base is the tri- 
angle ABO. 

Example. — A right prism with the triangular base A B Gis caused to 
revolve around its edge (7 by a force which acts uninterruptedly, so that 
at the end of the time t = lit has made ^ = f revolutions ; required not 
only the moment of this couple, but also the action of this motion upon 
the axis C. Let the base of this body be determined by the co-ordinates 

«i = lj5, 2/i = 0,5; «3 = 0,4, 2/2 = 1,0 feet, 
and let its length or height be ? = 2 feet, and its heaviness y == 30 pounds. 
From these data we calculate, first, the area of the base 

^x Vz ^2 Vx 



F = 



1,5 . 1,0 - 0,4 . 0,5 1,3 ^ ^^ 

-^ '—^ — '- = ^ = 0,65 square feet, 



and the mass of the whole body 

Fl y 



Now 



9 



t^-i 



= 0,031 . 0,65 . 2 . 30 = 1,209 pounds. 



+ x^X2 + ajg" = 2»^5 + ^'60 + 0,16 = 3,01 and 



Vx' +2/12/2 + y^' = 0,25 + 0,50 + 1,00 = 1,75, 



= 0,95914. 



hence the moment of inertia of the body is 

TF =(3,01 4- 1,75) 1^=- 4,76. ^-^^ 

In consequence of the constant action of the couj)le, the movement of 
rotation is uniformly accelerated, and consequently the angula/r velocit/y of 
the body at the end of the time i = 1 second is (see § 10) 



2 s 2.2 TT^ 



w ^= 



t t 

and the mechanical effect required is 



- = 31,416 feet, 



2 



§ 312.] THE CENTRIFUGAL FORCE OF RIGID BODIES. 633 

^ = 1 0)2 ^= 1 (31,416)' . 0,95914 = 473,3 foot-pounds. 
The angular acceleration is 

. = "- = M = 31,416 feet, 

V i. 

and therefore the moment of the couple 

Pa = K W= 31,416 . 0,95914 = 30,13 foot-pounds. 
The distances of the centre of gravity S of the base from the co-ordi- 
nate axes X X and T Y are 

^ = ^. +^^ = l-g + «- J = 0,6333 and 

O d 

. = H^ = ^^ = 0,5000, 
consequently the distance of the centre of gravity from the ams is 



C 8 = r = -^x' + y^ = 0,6511, 
Besides we have 

«* = 31,416* = 974090 and 
/c« = 31,416=^ = 987, 



whence 



Vw* + «^ = V975077 = 987,46, 
and the pressure upon tJie axis increases during the accelerated rotation from 

P=KMr = 31,416 . 1,209 . 0,6511 = 24,73 pounds 
to 

It = V«* + /c" . Mr = 987,46 . 1,209 . 0,651} = 777,33 pounds. 

If after one second of time the couple ceases to act, the motion of rota- 
tion of the body becomes uniform, and the pressure upon the axis from 
that moment consists only of the centrifugal force, which is 

N=d' Mr = 986,96 . 0,7872 = 776,94 pounds. 

The pressure upon the axis, which increases gradually from 24,73 to 
777,33 pounds, is in the beginning at right-angles to the central line of 
gravity G S^ but approaches more and more this line as the velocity 
increases, so that at the end of the time t — 1 second, it makes but an 
angle <p with that line, and this angle is determined by the expression 

-P 24 73 

tanr;. <!> = ^ = y^-^^- = 0,03183, 

for which o = 1" 40'. If the couple ceases to act, the direction of the 
axial force N = 776,94 pounds, coincides of course with the central line of 
gravity C S and revolves with this line in a circle. If instead of the couple 
a single force P acts with the arm a upon the body, another pressure equal 
to this force P must be added to the pressure on the axis. 



634 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 313. 



Fig. 520. 



§ 313. Centre of Percussion. — If the different portions 
Ml, Mci, etc., Fig. 520, of a system of revolving masses are not in 

one and the same plane, the 
directions of the forces 
§1 = 0)' Jfi a^i + K M^ yi, 
§2 = w' J/s iCa + « M^ y^, etc., 
no longer coincide with the 

co-ordinate axis X X, but 
lie in the co-ordinate plane 
X Z, and those of the forces 
i?j = w' ifi yi — « i/j iCi, 
7^2 = 0)'^ i/s y^— K. Jfs, x^, etc., 
no longer lie in the axis 
Y Y, but in the co-ordinate 
plane Y Z. The system of 
forces §1, Q.2, etc., and R^, E^, 
etc., give, according to § 305, 
the resultants 

Q = Qi -^ Q-2 + ' ' • and 
H =^ Hi -\- Re, + . , . 

Now since the lines of ap- 
plication U 'Q and F i? do 
not generally lie in the same 
plane, but cut the axis C Z 
of rotation at different points U and F, it is impossible to obtain a 
single resultant by combining them, but we can refer them to a 
fiingle force and a couple. The components are, of course, as above, 

§ = 0)' ( Jf, 2-1 + i/2 ^2 + . . .) + ^ (^1 yi + M^y^ + ...) 
— G)" Mx + K My 
and 

R = o)'{Miyi + M^y^ + ..,) — K (M^ Xi + M^x^ + , ..) 
— (^''My-\-KMx, 

Jf denoting the entire mass Mx -\- M<i + . , , and x and y the dis- 
tances of its centre of gravity 8 from the co-ordinate planes Y Z 
andXZ. 

Now if we put the distances of the masses Mi, M^, etc., from the 
plane of rotation X Y, which is perpendicular to the axis of rota- 
tion C Z, equal to Zi, z<i, etc., we obtain, as in § 305, the distances 
of the points of application U and F of the forces Q and R from 
the origin C. 




g 313,1 THE CENTRIFUGAL FOKCE OF RIGID BODIES. 635 

Q, z, + Qi z, -^ . . . 



It = 



_ (jS" (ifi x^ Zx + i/o x<, z^ + ...)+ /g {M\ y\ Zi + Mcj 1/2 Z.2 + . , .) 

"~ a77i^7^'i + i/"2^r+ . . .) + /^ (^1 2/1 + -^2 ^2 + . . .) 

and 

^1 2^1 + R2 2^0 + . . . 



f = 



Ri + ^2 + . . . 

_ w' (ifi ?/, 2;i -I- i¥2 2/2 ^2 -f- . . •) ^ '^ (-^1 ^1 ^1 + i/g i^2 2^2 + . • •) 
~ iS' {M^y, + M,y, + ...) — K [M, x, + M,x^ + . ..) * 

If the axis C Z is retained at two points A and B (the pillow 
blocks), which are at the distance C A = li and C B = k from 
the origin of co-ordinates, the force Q is decomposed into two com- 
ponents 

and the force R into the components 

Now the pressure upon the bearing A is 

Sind that upon the bearing B is 

If the acceleration of the rotation is produced not by a couple, 
whose moment is P a, but by a force P, whose arm is a, a third 
pressure equal to the force P is added to the two axial forces Q 
and R. If we cause this force P to act, at the distance F — a 
from the axis of rotation, parallel to the axis C Y and perpendicu- 
lar to the plane X Z, and if we assume that its line of application 
is at a distance C F — H = 1 from the co-ordinate plane X T, 
the force R only will be increased by an amount P, and the portion 
of it 1^1 at the bearing A will be increased by 



\k - ij 



P 
and the part Y^ at the bearing B by 



636 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 313, 



If MiX, -i- M,Xi + ,... = 0, 

Mi^i + M,y, + ,.. = 0, 
MiXiZi + M^Xc^Zci + , . . = and 
Ml yi Zi-\- M^y^z^ -^ . . . = 0, 

C ^ is a free axis, and not only the forces Q and R, but also 
their moments Q u and R v become = ; and we can, therefore, 
conclude that when a system of masses rotates about a free axis 
not only the centrifugal forces, but also the moments of inertia 
balance each other (compare § 309). 

Let us assume thit the system of masses is at rest, i.e., w = 0, 
or let us neglect thn action of the centrifugal force upon the axis 
of rotation, then we have more simply for the pressures in the axes 

Q= K My = ic (Ml yi + M^y., + . . .) and 

R ^ ^ fc M X = — K (Ml Xi + M^Xi + , . ,), and also 

Q u ••- « (Ml yi Zi + M^y^Z2 + . . .) and 

R ^ — — K (Ml Xi Zi + M^x^z^ -{- . . .). 

When the plane of X -^ is plane of symmetry and consequently 
9 ftiane o-f gravity, 

Miyi + i¥2 2/2 + ... = 

and 

MiyiZi + if2 2/2 2^2 + . . • = 0,' 

and, therefore, 

and also 

C ^ = 0. 

Now if we require that 
the force of rotation 

K W 



Fig. 521. 




F = 



a 



shall be counteracted by the 
force of inertia R, so that 
there shall be no action upon 
the axis of rotation, we must 
have 

P + R = 
and 

Ph -\- Rv = 0, 



^ 313.] 



THE CENTRIFUGAL FORCE OF RIGID BODIES. 



637 



and 



K W 

a 

K Wl 

a 



K {Ml Xi + M<iX2 + ., ,) = 



— K (Ml Xi Zi + M2X2Z2 -\- . . .) ^ 0, 



and consequently 



a = 



and 



Mx 



Ml Xi -\- Mi X2 + . , , 



'Ml Xx Zx + M^ x.-i Zi + . . . 



) 



a 



_ Moment of inertia 
Statical moment 

Ml Xi Zi + MiX<iZi + , .» 
Ml Xi -{- M^x^ + ,,, 



_ Moment of the centrifug al force 

~~ Statical moment. 

These co-ordinates determine a point 0, which is called the 
centre of percussion (Fr. centre de percussion ; Ger. Mittelpunkt 
des Stosses) ; for every force of impact P, whose direction passes 
through this point and is at right angles to the plane of symmetry 
X Z oi the body passing through the axis of rotation or fixed axis 
G Zy will be completely balanced, when the collision takes place, 
by the inertia of the mass, without producing any action upon the 
axis of the body. 

Example — 1) The moment of inertia of a straight line or rod G E^ 

Fig. 523, of uniform thickness throughout, which at one end G meets the 

axis C Z at a given angle Z G E^ when M is its mass 

and r the distance D E oi its other end from the axis 

of rotation, is 

W= MTc^ =^ Mr^ (see § 386), 

and, on the contrary, the statical moment is 

Mx = ^ Mr^ 

and finally the moment of the centrifugal force, since, 
if Ti denotes the projection G D oi the length G E oi the 
rod on the axis of rotation G Z, vfe have 



Fig. 533. 




or 



is 



G 0^ _ z, 
0^ M^ ~~ x^ 



h 



^1 «i «i = - ^1 «l^ ^2 ^z^z = - ^z '^2^ etc., 



r 



Therefore, the co-ordinates of the centre of percussion of this rod are 



JVj «! «! +M^ ajg ^2 +.-. = - (^1 a^i^ + -a/'g ajg^ +...) = ^ . I Mr^=^Mhr. 



determined by the formulas 



638 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 313. 



and 



F = a = 



G F ^ l = 



Moment of inertia 



J- Mr"" 



Statical moment 
Moment of centrifugal force 



= #r 



^ Mr 



= 1^ 



Fig. 523. 



Statical moment 

and this centre is situated at f of the length G B of the rod from the end 
G and ^ of the same from the end F. 

2) The moment of inertia of a surface A B G, Fig. 523, whose form is a 
right-angled triangle, which turns around its base G A, 
is, when we denote the mass by M and its base and 
perpendicular G A and G Bhy h and r, 

Ti r^ Ji r r"^ 
T=-^= -^ .- = \Mr^ (see § 229), 

and its statical moment, since the centre of gravity S 

r 
is at a distance - from the axis (7 ^, is 
o 




Mx = 



Mr 
~3~' 



consequently the distance of the centre of percussion 
of this surface from this axis is 



F = a = 



^Mr"" 
^Mr 



= 4 r. 



For an element K L of the triangle, whose shape is that of a strip, 
whose length is x and whose width is -, and which is situated at a dis- 



n 



tance G K = z from the apex (7, the moment of the centrifugal force is 



Mxz=~x.^xz, 



n 



X r 
or, smce - = t, or « 



Mx z = X 



h 



n 



0" 



' \^/' \^/ * ' * \nj 



Substituting for z successively the values 1 

and adding the values thus obtained for Mxz,we have the total moment 
of the centrifugal forces 

M,x,z,-^M,x,z,^... = ^\ Q\l^ + 2^ + 3^ +. . . . + ^^) Q' 

= lMrh, 

and, therefore, the distance of the centre of percussion from the comer 

(7 is 

iMrh 



GF= I = 



^Mr 



= f A. 



§ 314.] 



THE ACTION OF GRAVITY, ETC. 



639 



CHAPTEU III. 

OF THE ACTION OF GRAVITY^ UPON THE MOTION OF BODIES 

IN PRESCRIBED PATHS. 



Fig. 524. 



§ 314. Sliding upon an Inclined Plane. — A heavy body can 
be hindered in many ways from falling freely. We will, however, 
consider but two cases here, viz., the case of a body supported by 
an inclined plane and the case of a body movable around a hori- 
zontal axis. In both cases the paths of the bodies are contained in 
a vertical plane. If a body lies upon an inclined plane, its weight 
is decomposed into two components, one of which is normal to the 
plane and is counteracted by it, and the other is parallel to the 
plane and acts upon the body as a motive force. Let G be the 
weight of the body A B C D, Fig. 524, and a angle of inclination of 

the inclined plane F H Rio the^ 

horizon, according to § 146 t\ 

normal force is 

JSf = G COS. a, 

and the motive force is 
P = G sin. a. 

The motion of the body can 
be either a sliding or a rolling 
one. Let us consider the former 
ease first. In this case all the 
parts of the body participate equally in its motion, and have there- 
fore a common acceleration jo, determined by the well-known formula 
force P G sin. a 
^ mass' M G ^ ^ ' 

hence p : g = sin. a : 1, 

I.E., the acceleration of a body upon an inclined plane is to the accel- 
eration of gravity as the sine of the angle of inclination of the plane 
is to unity. But on account of the friction this formula is seldom 
sufficiently accurate. It is, therefore, very often necessary in prac- 
tice to take the friction into consideration. 

If a body moves upon a curved surface the acceleration ig 




640 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 315. 



variable, and is in eyery point equal to the acceleration correspond- 
ing to the plane, which is tangent to the curved surface at that 
point. 

§ 315. If a body slides down an inclined plane without fric- 
tion and its initial velocity is = 0, then, according to § 11, the 
final velocity after t seconds is 

V — g sin. a .t = 32,2 sin. a , t feet = 9,81 sin. a . t meters, 
and the space described is 
s = \ g sin. a . f =^ 16,1 sin. a . f feet = 4,905 sin. a . f meters. 

When a body falls freely Vi = g t and Si = ^ g f , and we can 
therefore put 

V : Vi = s : Si = sin. a : 1, 

I.E., the final velocity and the space described by a body sliding upon 
the inclined plane are to the velocity and the space described by a 
body falling freely as the sine of the angle of inclination of the plane 
is to unity. 

In the right-angled triangle F G H, Fig. 525, whose h3rpothenuse 
Fig 525 F Gh vertical, the base is F II = F G sin. FG H — 
^ F G sin. F II E = F G sin. a, when a denotes the 
inclination of the base to the horizon, and therefore 

F ff: F G = sin. a : 1 ; 

the body, therefore, describes the vertical hypothenuse 
F G and the inclined base F H in the same time. 
Hence the space described by a body upon an inclined 
plane in the time, in which, if falling freely, it would 
describe a given space, can be found by construction. 

Since all the angles F Hi G, F H^ G, etc., inscribed in a semi- 
circle F Hi G, Fig. 526, are right angles, the semicircle subtended 

hj F G will cut off from all inclined 
planes beginning at F the distances 
F Hi, F H^, etc., described simultane- 
ously with the diameter. For this rea- 
son we say that the chords or diameter 
of a circle are described simultaneously 
or isochronously. This is true not only 
when the chords, as, e.g., F Hi, F H^, 
etc., begin at the highest point F, but 
also when the chords, as, e.g., Ki G, Kc^ G, 
etc., end at its lowest point G', for we 




Fig. 536. 




§316.] THE ACTION OF GRAVITY, ETC. 641 

can draw through F the chords F Kx, F K^, etc., which have the 
same length and position as the chords G H^, G H<i, etc. 

§ 316. From the equation 

s — ^r— = t: — —. for the space described, 

2^; 2g . sin. a 

we obtain 

s sin. a = 7i— , and inversely, 
2g ^ 



V = V % g s sin. a. 

Now s sin. a is the height F R (Fig. 527) of the inclined plane 
or the vertical projection h of the space F H = s. If, therefore, 
several bodies, whose initial velocities are = 0, descend inclined 
Fig. 527. planes F H, F H^, etc., of different inclina- 

tions, but of the same height, their final 
velocity will be the same and equal to that 
acquired by a body falling freely through 
the distance F R (compare § 43 and § 84). 
From the equation s = ^ g sin. a . f we 
obtain the formula for the time 




, _ a/ ^~5 _ 1 a/2 s sin. a __ 1 /2 h 

^ g sin. a sin. a g sin. a' ^ g ' 

K a body falls freely through the height F R = h, the time ig 

ti — y — , whence 

9 
t:tx = l'. sin. a = s:h = F H: F R. 

Hie time required hy a body to descend an inclined plane is to the 
time of falling freely through the height of this plane as the length 
of the plane is to its height. 

Example— 1) The top F of an inclined plane F H^ Fig. 528, is given, 
and we are required to determine the other extremity H^ which is situated 
in such a position upon a line A B that a body descending the plane will 
reach this line in the shortest time. If through F we draw the horizontal 
line F O until it cuts A J5, and make G H = G F, we obtain in H the 
point required, and in Fff the plane of the quickest descent ; for if we 
pass through F and H& circle, to which the lines F H and G H are tan- 
41 . 



642 



GENERAL PRINCIPLES OF MECHANICS. 



[% 317. 



gents, the chords F K^, F ^g,etc., described simultaneously, are shorter 

than the lines F ff^, F Ho, etc., drawn from i^to 




■s? 



the line A B; consequently the time required to 
descend this chord is less than that required to 
descend these lines, and the inclined plane F H^ 
which coincides with that chord, is the plane of 
quickest descent. 

2) Required the inclination of the inclined 
plane F H^ Fig. 527, which a body will descend 
in the same time as it will fall freely through the 
height F B and move with the acquired velocity 
upon a horizontal plane to H. The time required 
to fall through the vertical distance F B = his 



«i = 



2 h 




Q 



, and the velocity acquired is 



« = ^ "i gTi, 

If no velocity is lost in passing from the vertical to the horizontal mo- 
tion, which is the case when the corner B is rounded off, the space B H 
= Ji cotg. a will be described uniformly and in the time 

h cotg. a Ji cotg. a /"o 

^cotg. aA/^_ 



h = 



2 h 
9 



^ ^/ 2 gJi 

The time in which a body will descend the inclined plane is 

sin. a ' g 
Now if we put t = t^ + t^, we obtain the equation of condition 



= 1+1^ cotg. a or 



ta?ig. a 



tang, a + ^. 



Resolving this equation, we obtain tang, a = ^. In the corresponding 
inclined plane the height is to the base is to the length as 3 is to 4 is to 
5, and the angle of inclination is a = 36° 52' 11". 

3) The time in which a body will slide down an inclined plane, whose 
base is a, is 



f a sin. a f a sm. a cos. a r c 



4 a 



g sm. a f g sm. a cos. a f g sin. 2 a ' 
this is a minimum when sin. 2 a is a maximum, i.e. = 1 ; then -2 a° = 90 
or a° = 45°. "Water flows quickest down roofs whose pitch is 45°. 

§ 317. If the initial velocity of a body upon an inclined plane 
is c, we must employ the formula found in § 13 and § 14 ; hence, 
when a body ascends an inclined plane, we have the velocity 

V = c — g sin, a . t, 
and the space described 



§318.] THE ACTION OF GRAVITY, ETC. 643 

s = c t — I g sin. a . f, 
and for a body descending the inclined plane we mnst put 

V = c + g sin. a . t and s = c t -{- ^ g sin. a . f. 
In both cases, however, the following formula 

v' - c' . , v' - C' v' c" 



s = : , or s sin. a = h 




2 g sin. a' 2g 2 g 2 g 

is applicable. 

The vertical projection {h) of the space {s) described upon the in- 
clined plane is ahuays equal to the difference of the heights due to tJie 
velocities. 

When two inclined planes F G Q and O H R, Fig. 529, meet in 
a rounded edge, a body descending the plane will experience no 

impact in passing from one to the other ; 
^^' '^ ■ ^ hence, if we have such a combination of 

planes, there will be no loss of velocity, 
and the following rule will be applicable 
to the case of a body descending these 
planes: height of fall equal to height due 
to velocity. We can easily understand that 
when a body ascends or descends a series of such planes or a curved 
line or surface, its motion will take place according to the same law. 

Example — 1) A body ascends, with an initial velocity of 21 feet, an 
inclined plane, the inclination of which is 22°. What is its velocity and 
what is the space described after 1^ seconds ? 

The velocity is 

>o = ^1 — 32,2 sin. 22° . 1,5 = 21 — 32,2 . 0,3746 . 1,5 = 21 - 18,09 

= 2,91 feet, 

and the space is 

c + V 21 + 2,91 ^ 23,91 . 3 , 
8 = -^ . t = ^— . f = -^ = 17,93 feet. 

2) How high will a body, whose initial velocity is 36 feet, rise upon a 
plane inclined at 48° to the horizon ? The vertical height is 

h = ~ = 0,0155 . v^ = 0,0155 . 36^ = 20,088 feet, 

2<7 ' ' ' ' 

and therefore the entire space described upon the inclined plane is 
_ h 20,088 



sin. a sill. 48° 
and the time required to describe it is 

2.8 2 . 27,031 27,031 



= 27,031 feet, 

= 1,5 seconds. 



D 36 18 

§ 318. Sliding upon an Inclined Plane when the Fric- 
tion is taken into Consideration. — The sliding friction has 



eu 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 818. 



great influence upon the ascent or descent of a body upon an in- 
clined plane. From the weight G of the body and from the angle 
of inclination a we obtain the normal pressure 

JSf = G- COS. a, 
and consequently the friction 

F = (p JV = (f) G COS. a. 
If we subtract the latter from the force Fi = G sin. a, with which 
the gravity pulls it down the plane, there remains the motive force 

F = G sm. a — (f) G cos. a, 
and we have for acceleration of a body moving down the inclined 
plane 

force (G sin. a — (b G cos. a\ , , ^ . 

^ = S^s = I G i ^ = ('*'*• " - ^ '''■ ") ^- 

For a body ascending an inclined plane the motive force is neg- 
ative and = 6^ sin. a + cj) . G cos. a, and the acceleration p is also 
negative and = — {sin. a + (p cos. a) g. 

If two bodies placed upon two different inclined planes F G and 

F H, Fig. 530, are united by a perfectly 
flexible cord, which passes over a pulley 
C, it is possible that one of the bodies 
will descend and raise the other. De- 
noting the weight of these bodies by G 
and G-i, and the angles of inclination 
of the inclined planes, upon which they 
rest, by a and a,, and assuming that G 
descends and draws up Gi, we obtain the motive force 

F — G sin. a — Gx sin. a^ — (ji G cos. a — (p Gi cos. «! 
= G {sin. a — (f) COS. a) — Gi {sin. ai + cos. ai), 
and the mass moved 

,-. G + Gx 

M := -^ 

9 
and therefore the acceleration with which G descends and Gi 

ascends is 

G {sin. a — (j) cos. a) — Gx {sin. ax -\- (p cos. a^) 



Fig. 530. 




P 



9- 



G + Gx 

Since the friction, •^Yhich is a resistance, cannot produce mo- 
tion, we must have, if G descends and Gx ascends, 

G {sin. a — (p cos. a) > Gx [sin. a^ + (p cos. Oj), or 

G sin. a^ + (p COS. a^ ^ -^ '^^'^- (^i + P) 

Gx sin. a — (p cos. a ' ' ' Gx sin. {a — p)' 



§318.] THE ACTION OF GRAVITY, ETC. ' 645 

If, on the contrary, G^ descends and G ascends, we must have 

Gx ^ sin. a + (h COS. a 

7T > -■ T J or 

G sm. tti — COS. ttj 

G sin. ttj — COS. a, ^ / '^*^' (^i ~ P) 

Gx ^ sin. a + COS. a ' ' ' Gx sin. {a + p)' 

As long as the ratio -^ is within the limits 

si7i. a, + COS. a, ^ sin. a. — cb cos. a^ 

—. — - — —_ and —. — ^ , or 

sm. a — (j) COS. a sin. a -\- (p cos. a 

sin. («] 4- p) T sin. (a^ — p) 

—. — 7 ^ and -—. — ) -, 

sm. (a — p) sm. [a + p) 

the friction will prevent any motion. 

Example — 1) A sled slides down an inclined plane covered with snow, 
150 feet long and inclined at an angle of 20 degrees, and on arriving at the 
bottom it slides forward upon a horizontal plane until the friction brings 
It to rest. If the coefficient of friction between the snow and the sled is 
= 0,03, what space will the sled describe upon the horizontal plane (the 
resistance of the air being neglected) ? 
The acceleration of the sled is 

p = [sin. a — (j) COS. a) g = (sin. 20° — 0,03 . eos. 20°) . 32,2 
= (0,3420 - 0,03 . 0,9397). 32,2 = 0,3138 . 32,2 = 10,104 feet, 

and therefore its velocity on arriving at the bottom of the inclined plane is 

V = V2j9s = V2. 10,104 . 150 = V3()31^2 = 55,06 feet. 

Upon the horizontal plane the acceleration is 

p^ = —(pg = — 0,03 . 82,2 = — 0,966 feet, 

and therefore the space described is 

«" 3031,2 

'^=¥J-g = T2^ = ''''''''■ 

The time required to slide down the inclined plane is 

2 s 300 
t = — = ^■, ^„ = 5,45 seconds : 
V 55,06 ' ' 

that required to slide along on the horizontal plane is 

,:. ^o^ .2 s. 3138 

Fig. 531. t, —- — ^ = = 57 seconds 

^ V 55,06 ««^"iiufc', 

and therefore the duration of the entire journey is 
t + t^ = 62,45 seconds = 1 minute 2,45 seconds. 
2) A bucket K, Fig. 531, which, when filled, weighs 
250 pounds, is drawn up a plane, 70 feet long and in- 
clined at an angle of 50°, by a weight G = 260 ; what 
time will be required when the coefficient of the fric- 
tion of the bucket upon the floor is 0,36 ? 




646 GENERAL PRINCIPLES OP MECHANICS. [§819. 

The motive force is 

z= G — {sin. a + (pcos. a) K = 260 — (sin. 50° + 0,36 cos. 50°) . 250 

= 260 - 0,9974 . 250 = 10,6 pounds, 

and therefore the acceleration is 

10,6 10,6 

^=^250T260= 5iO = ^'^^^^^'"*5 

the time of the motion is 

t = y — = y Q-0208 ~ ^^^ = ^^'^^ s^c- = 1 ^^' 2^ sec, 

and the final velocity 

2 s 140 , ^^ , 

^ = T = ¥2-=1'^^^^^*- 

§ 319. Rolling Motion upon an Inclined Plane. — When 
a wagon runs down an inclined plane, it is the friction on the axle 
which offers the principal resistance to the acceleration. If G is 
the weight of the wagon, r the radius of the axle and a that of the 
wheel, we have 

L. — ^ ^ r_. Q COS. a, 
a a 

and therefore the acceleration 



/ . r \ 

'p = \8%n. a — - — COS. a) g. 




If a round tody A B, as, e.g., a cylinder or a sphere, etc., rolls 
down an inclined plane F H, Fig. 532, we have at the same time a 

motion of translation and of rotation. As 
the acceleration of translation j^ is generally 
equal to that of rotation (§ 169), if we put 
the moment of inertia of the rotating body 
=^ G h^ and the radius C A oi rotation = a, 

we obtain for the force A K — K, with 
which the roller (in consequence of the mu- 
tual penetration of its surface and that of 
the inclined plane) is set in rotation, 

^ G¥ 

But the force K opposes the force G sin. o, which tends to 
cause the body to slide down the plane, and therefore the motive 
force for the motion of translation is 

P = G sin. a — K, 

and its acceleration is 

G sin. a — K 
p = g . g 



§319.] THE ACTION OF GRAVITY, ETC. 647 

Eliminating K from the two equations, we obtain 

Q p — G g sm. a ^— .p, 

and consequently the required acceleration 

a sin. a 

For a homogeneous cylinder Ic" — ^ c^ (§ 288), and therefore 

a sin. a ^ 

but for a S'pliere k'' = % a" (§ 290), and therefore 

a sin. a , 
P = ^ , 2^ = i9 sin.a', 

5 

the acceleration of a rolHng cylinder is but | and that of a rolling 
sphere is but f as great as that of a body sliding without friction. 
The force which produces the rotation is 

-r^ _ g sin. a Gh"^ __ Gk"^ sin. a 

~ _, Jc'' ' g a^ ~ a' + F 

1 + ^ 
a' 

As lon^ as this force is less than the sliding friction G cos. a, 
so long will the body descend the plane with a perfect rolling 
motion. But if 

Ky (p G cos. a, 1.^., if tang, a > (/> (1 + -^j, 

the friction is no longer sufficient to impart a velocity of rotation 

equal to that of translation ; the acceleration of translation 

becomes, as in the case of sliding friction, 

G sin. a — (h G cos. a , . , 

p = ^ . g = (stn, a — (/) cos. a) g, 

and that of rotation 

G COS. a a^ 

P' = ^WV^' • 9 = t' 1^' 9 oos. a. 

If the Aveight of a wagon is G, the radius of its wheels a and 
their moment of inertia G ¥, we will have 

r 
-, , , G sin. a — (p ~ G cos. a — K 

h = p — and p — . fir, 

^ g a" ^ G ^ 



648 GENERAL PRINCIPLES OF MECHANICS. [§330. 



!9 



T 

g {sin, a — (ji - cos, a) 
(t 



Example — 1) A wagon, which, when loaded, weighs 3600 pounds and 
whose wheels are 4 feet high and have a moment of inertia of 2000 foot- 
pounds, rolls down a plane whose inclination is 12° ; required the accelera- 
tion, when the coefficient of friction upon the axles is ^ = 0,15 and the 
thickness of the axles is 2 r = 3 inches. 

Here we have 

and therefore the required acceleration is 

_ 82,2 {sin. 12°-0,0094 . eos. 12°) _ 32,2 . (0,2079 — 0,0094 . 0,978) 
^~" 1 -f 0,139 ~ 1,139 

2) With what acceleration will a massive roller roll down a plane whose 
angle of inclination is a = 40° ? 

If the coefficient of sliding friction of the roller upon the plane is 
^ = 0,24, we have 

^(l + -^) =0,24(1 + 2) =0,72. 

Now tang. 40° = 0,839, and tang, a is therefore greater than <^ ( 1 + p ), 

and the acceleration of the rolling motion is smaller than that of the mo- 
tion of translation. 

The latter is 

p = {sin. a - <pcos.a)g = (0,648 - 0,24 . 0,7660) . 32,2 = 0,459. 32,2 
= 14,78 -feet, and the former is 

p^ = 0,24 . 2 . 32,20 cos. 40° = 15,456 . 0,776 = 11,99 feet. 

§ 320. The Circular Pendulum. — A body suspended from 
a horizontal axis is in equilibrium as long as its centre of gravity 
is vertically under this axis ; but if we move the centre of gravity 
out of the vertical plane containing the axis and abandon the body 
to itself, it assumes an oscillating or vibrating motion (Fr. oscilla- 
tion, Ger. Schwingende Bewegung), i.e., a reciprocating motion in 
a circle. A body oscillating about a horizontal axis is called a 
pendidiim (Fr. pendule, Ger. Pendel or Kreispendel). If the 
oscillating body is a material point, and if it is connected with the 
axis of rotation by a line without weight, we have a simple or 
theoretical pendulum (Fr. p. simple, Ger. einfaches or mathema- 



§321.] THE ACTION OF GRAVITY, ETC, 649 

tisclies P.) ; but if the pendulum consists of a body or of several 
bodies of finite dimensions, it is called a compound pendulum (Fr, 
pendule compose, Ger. zuzammengeseztes, physisches or materielles 
Pendel). Such a pendulum can be considered as a rigid combina- 
tion of a number of simple pendulums, oscillating around a 
common axis. The simple pendulum has no real existence, but it 
is of great use in discussing the theory of the compound pendu- 
lum, which can be deduced from that of the simple one. If the 
pendulum, which is suspended in C, Fig. 533, is moyed from its 
vertical position C if to the position C A and left to itself, by 
virtue of its weight it will return towards C M with an accelerated 

motion, and it will arrive at the point M 
with a velocity, the height due to which is 
equal to D M. In consequence of this 
velocity it describes upon the other side 
the arc MB = MA, and rises to the 
height D M. It falls back again from B 
to M and A and continues to move back- 
wards and forwards in the arc A B. If we 
could do away with the friction on the 
axis and the resistance of the air, this 
oscillating motion of the pendulum would continue forever ; but 
since these resistances can never be entirely removed, the arc in 
which the oscillation takes place will gradually decrease until the 
pendulum comes to rest. 

The motion of the pendulum from ^ to ^ is called an oscilla- 
tion (Fr. oscillation, Ger. Schwung or Pendelschlag), the arc A B, the 
amplitude (Fr. amplitude, Ger. Swingungsbogen), and the angle 
measured by half the amplitude is called the angle of displacement. 
The time in which the pendulum makes an oscillation is called the 
time, duration, or period of an oscillation (Fr. duree d'une oscilla- 
tion, Ger. Schwingungszeit or Schwingungsdauer). 

§ 321. Theory of the Simple Pendulum. — In consequence 
of the frequent use of the pendulum in common life, viz. for clocks, 
it is important to know the duration of an oscillation ; its demon- 
stration is therefore one of the most important problems in 
Mechanics. To solve this problem, let us put the length of the 
pendulum A C = M C = r, Fig. 534, and the height of rise and 
fall during an oscillation M D — h. Assuming that the pendulum 




650 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 321. 



has fallen from A to G, and making the vertical height D H oi fall 
corresponding to this motion = x, we have the velocity acquired 
at G 

V = V2 g X, 

and the element of time, during which 
the element of its path G Kis described, 
GK G K 



Fm. 534. 




T = 



^ \^1g X 

If we describe from the middle of 
MD^h with the radius OM = 0D = 
^ li the semicircle M ND,wq can cut from 
the latter an elementary arc N P, which 
will hare the same altitude P Q = 
K L = R H 2i^ G K, and whose relation 
to the latter can be very simply ex- 
pressed. In consequence of the sim- 
ilarity of the triangles G K L and G G H ^q have 

GK _ C G 
K L~ ~GH' 

and in consequence of the similarity of the triangles N P Q and 
NH 

NP _ ON 

P Q ~ NH' 
dividing the first of these proportions by the second and remem- 
bering that K L = P Q, we obtain the ratio of the above elements 
of the arcs 

G K _ G G .NH 

N P~ G H. N' 
Erom a well-known property of the circle we have 

GlP = MH{2 CM- MH)sind NH' ^ M H . D H, 

whence it follows that 

r Vx 



G K 



G G .VDH 



NP ON.V^cM-MH -z^V^r - {h-xi 
and the time required to describe an element of the path is 
r Vx N P 2 r 



r = 



i h i/2 r - (h-x) ' V'igx 
_ Jr NP 



^V2g[2r - (h - x)] 



NP 



9 



h\/i __ 



li -~x 
2r 



^322.] THE ACTION OF GRAVITY, ETC. 651 

Generally in practice the angle of displacement is small, and 

then 7^-, TT- and -t, — are such small quantities, that we can 
2 r 2 r 2 r 

neglect them and their higher powers and put 



■/^ 



NP 



cj " n 

The duration of a semi-oscillation or the time within which the 
pendulum describes the arc A M \^ equal to the sum of all the 
elements of the time corresponding to the elements G K or JSF P. 

Now since ^ . y - is a constant factor, we can put the sum equal 
h g 

to T y - times the sum of all the elements forming the semi- 
h ^ g ^ ^ 

circle D N M, i.e., = - y - times the semicircle l-^-)? or 

_ 1 .,^ IT li __ TT ./r 
^'~h^g'^~2'^g 

The same time is required by the pendulum for its ascent; for 
the velocities are the same but opposite in direction, hence the 
duration of a complete oscillation is double the latter, or 

t = 2ty = 7T i/-. 

^ 9 

(§ 322 ) More Exact Formula for the Duration of an 
Oscillation of the Circular Pendulum. — In order to determine 
the duration of an oscillation with greater precision, as is some- 
times necessary, when angles of displacement are large, we caa 
transform the equation 

' (■ - -^r 



\/l- ^-^ 



2r 
into the series 



^'^ ' ~2V "^ ^•r2^/ + •••' 

and then we have the time in which an element of the path is 
described 

r, ^ , li-x ^ . (Ji-xW -\./r NP 



652 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 323. 



Putting the central angle D W = (p% or the arc 
FiG.535. DN=D0..p = ^4-, 




we obtain the height 
Mir= h - X = M 



HO=l 



+ ^ COS. (/) = (! + COS. (p) I ; 
and therefore the element of time 



T = [l + 1 . (1 + COS. (/)) 



h 



4 r 
NP 



or, smce 

(1 + COS. 0)'^ = 1 + 2 COS. + {cos. 0)' = 1 + 2 COS. </> -t 

= 1 + 2 cos. + 3 COS. 2 0, 
r = [l + i (1 + COS. 0) A + 3(3 + 2 ^a.. d) 

5^ 



9 h ' 
1 + COS. 2 



2 



+ icos.2 0)(A)V...]|/|. 



A 



Now the sum of all the elements iV P is = the arc D N P = 



N P COS. <p \^ — N Q and the sum of all the iV^ § is = the 



h 

2 ' 

ordinate N H — - sin. and also the sum of all the ^ — '- — 

2 h 

is = sin. 2 0, therefore the time required to describe the arc AGi% 



+ T^i 



§322.] THE ACTION OF GRAVITY, ETC. 653 

The time required to describe the arc A M is, since we have 
here (f) = n, sin. (p = sin. n and sin. 2 = sin. 2 rr = 0, 

As the velocity decreases in the same manner, when the pen- 
dulum ascends on the other side, as it increased during the descent, 
the time required for describing the entire arc or the duration of 
the complete oscillation is 



(lii^J (rfh ■ • ^'^ ^l- 



+ 

a 

If the pendulum oscillates in a semicircle, we hare A = r, and 
consequently the duration of an oscillation is 

In the most cases in practice the amplitude of the oscillations 
is much less than a semicircle, and the formula 

\ 8 r/ g 

is sufficiently accurate. 

If the angle of displacement be denoted by a, we have cos, a = 

= 1 or- = l — COS. a, and therefore 

r r r 



h _ ^ 1 — COS. a _ J 
8T ~ ^ • 2 ~ 4- 



(sin. l)\ 



from the latter formula we can determine the correction to be 
applied for any given amplitude. If, for example, this angle if 
a =: 15°, we have 






00426. 



and, on the contrary, for a = 5° 

~ = 0,00047 ; 
8 r 

for this last amplitude the duration of an oscillation is 

t = 1,00047 . 77 i/i. 

9 



654 GENERAL PRINCIPLES OF MECHANICS. [§333. 

Consequently if the amplitude is less than 6% we can put with 
sufficient accuracy the duration of an oscillation 

t = rr i/- = -^Vr = 0,554 Vr. 
9 Vg 

§ 323. Length of the Pendulum. — Since in the formula 

9 
the angle of displacement does not appear, it follows that the 

duration of small oscillations of a pendulum does not depend upon 
this angle, and that pendulums of the same lengths, when their 
amplitudes, although different, are small, oscillate isochronally or 
have the same duration of oscillation. A pendulum, when its am- 
plitude is 4 degrees, make an oscillation in (almost) the same time 
as when it is 1 degree. 

If we compare the duration t of an oscillation with the time tx 
of the free fall, we find the following relation. The time required 
by a body to fall freely a distance r is 

^9 ff 

hence 

the duration of an oscillation of a pendulum is to the time required 
by a body to fall freely a distance equal to the length of the pen- 
dulum as the number tt is to the square root of 2. The time re- 
quired to fall the distance 2 r is 



w/2.2r . Jt 



t, = y -^-^^ = 2 1/ — 
9 9 

therefore the duration of an oscillation is to the time required to fall 

a height equal to twice the length of the pendulum as n is to 2. 

If we put the durations of the oscillations of two pendulums, 

whose lengths are r and ri, equal to t and ti, we obtain 

t : ti = 'S^ r : V r^. 
When the acceleration of gravity is the same, the durations of the 
oscillations are 2^roportional to the square roots of the lengths of the 
pendulums. Now if n is the number of oscillations made by one 
pendulum in a certain time, as, e.g., in a minute, and n^ the num- 
ber made in the same time by another pendulum, we have 

^/~V: 4^ 7*1 ra - : — 
n Ux 



§324.] THE ACTION OF GRAVITY, ETC. 655 

and inversely n\nx — VTl : 4^ r , 

I.E. the number of oscillations is inversely proportional to the square 
root of the length of the ]p&ndulum. A pendulum four times as 
long as another makes but one-half as many oscillations in the 
same time. 

A pendulum is called a second pendulum (Fr. pendule a seconde, 
Ger. Secundenpendel), when the duration of its oscillation is a 




second. Substitutinsr in the formula ^ = tt 4/ — , ^ = 1, we obtain 

q 
the length of the second pendulum r = —^ ; for English system 

of measures 

r .= 3,26255 feet = 39,1506 inches, 
and for the metrical system 

r = 0,9938 metres. 

By inverting the formula t = re y — , we obtain g = (-) ^j ^7 

means of which we can deduce from the length r of the pendulum 
and the duration t of its oscillation the acceleration g of gravity. 
We can determine the value of g more simply and more accurately 
in this manner than with Atwood's machine. 

Remakk. — By observations upon the pendulum, the decrease of the force 
of gravity, as we proceed from the equator to the poles, has been proved,^ 
and its intensity determined. This diminution is caused by the centrifugal 
force arising from the daily revolution of the earth upon its axis, and also 
by the increase of the radius of the earth from the poles to the equator. 
The centrifugal force diminishes the action of gravity at the equator ^-^-^^ of 
its value (§ 303), while at the poles the action of the centrifugal force is null. 
By observation upon the pendulum we can determine the acceleration of 
gravity at the place of observation. This acceleration, when (3 denotes the 
latitude of the place, is 

g = 9,8056 (1 - 0,00259 cos. 3 (3) metres; 
therefore at the equator, where 3 = and cos.2(3 = 1, we have, 

g = 9,8056 (1 - 0,00359) = 9,780 metres, 
and at the poles, where J3 = 90°, cos. 2 (3 = cos. 180° = — 1, 

g = 9,8056 . 1,00359 = 9,831 metres. 
Upon mountains g is smaller than at the level of the sea. 

§ 324. Cycloid. — We can put a body in oscillation or cause it 
to assume a reciprocating motion in an infinite number of ways. 
Any body moving in such a manner is called a pendulum. We 
distinguish several kinds of pendulums, as, for example, the circu-- 
lar pendulum, which we have just discussed, the cycloidal pendulum^ 
where the body, by virtue of its weight, swings backwards and for- 



656 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 32o. 




wards in a cycloid, and the torsion penduhim or torsion balance, 
where a body oscillates in consequence of the torsion of a string or 
wire, etc. We will here discuss only the cycloidal pendulum. 

The cycloid (Fr. cycloide, Ger. Cycloide) A P, D, Fig. 536, is a 

curve described by a 
Fm. 536. point ^ of a circle 

A P B, which rolls 
upon a straight line 
B D. If this gene- 
rating circle rolls for- 
ward the distance 
B B, = C C, and 
comes into the posi- 
tion Ai B^y it turns 
through the arc ^4 P 
= Ai Pi = B B^ = P Pi, and the ordinate M Pj corresponding 
to any abscissa A M is = ordinate JIf P of the circle plus the arc 
A P, which the circle has turned. In this rolling the generating 
circle turns always upon its point of tangency to the base line B D ; 
if it is in -4 1 P„ it turns about By, and thus describes the element 
Pi Q] of an arc of the cycloid ; consequently the chord B^ Pj gives 
the direction of the normal and the chord A-^ P^ that of the tangent 
Pi T at the point Pj of the cycloid. The prolongation P § of the 
chord A P reaching to the ordinate Qi is equal to the element 
Pi Qi of the cycloid ; since the space P R due to the motion of ro- 
tation is equal to that R Q due to the motion of translation, P § is 
the base of an isosceles triangle, and is equal to twice the line P N, 
which is cut off by the perpendicular R iV; P i\^is finally the dif- 
ference of the two neighboring chords A R and A P, and conse- 
quently the element Pj Qi of the cycloid is equal to twice the 
difference (^ P — ^ P) of the chords. Since the successive ele- 
ments of the cycloid compose the arc A Pi, and the sum of the 
differences of the chords the entire chord A P, we have the length 
of the arc A Pi of the cycloid equal to twice the chord ^ P of the 
generating circle. The diameter of the circle is the chord corre- 
sponding to the semi-cycloid, and the length of the semi-cycloid is 
therefore twice the diameter (2 ^ P) of the generating circle. 

§ 325. Cycloidal Pendulum. — From the properties of the 
cycloid, found in the foregoing paragraph, we can easily deduce the 
theory of the cycloidal pendulum, or the formula for the duration 
of an oscillation of a body vibrating in the arc of a cycloid. Let 



335.J 



THE ACTION OF GRAVITY, ETC. 



657 



A K M^ Fig. 537, be half the arc of the cycloid, in which a body 
oscillates, and M E the generating circle, whose radius is C U = 



Fig. 537. 




C M = r. If the body has described the arc A G or fallen from 
the heigTit D H = x (compare § 321), it has attained the velocity 

V = V2 gx, with which it describes the element G Xot the arc in 

the time 

GX GK 

T = : 



^ V2gx 

In consequence of the similarity of the triangles G L ^and FHM, 

we have 

G K _ F M 

KL~ MW 



or, since F M"" = MR. ME, 

G K VM H.M'E VWW 



KL 



M H 



V~mh' 

and in consequence of the similarity of the triangles N P Q and 
ONE 

JSTP _ 01^ 

P Q~ NIT 

or, since WIP = M H , D H, 
NP _ ON 

P Q~ VMH .DH* 
Now K L — P Q^ hence by division we have 

G K _ VWe VMH .D H _ VWeTWH 
NP 



VMH 



ON 



N 



h 



or, since N, half the height fallen through, = ^, M E = 2 r md. 
DR=x, 



GK V2rx 2 V2Vx 



NP 



hh 



h 



42 



658 



GENERAL PRINCIPLES OF MECHANICS. 



[§325. 



If we substitute G K 
G K 



2V2rx 



T = 



V2 gx 



h 
we obtain 



iV^ P in the formula 



2 \2 rx 



,NP = 



9 



NP, 



V2 g x.h 

The time required to fall from A to M is the sum of all the 
values of r, obtained by substituting for JV P all the divisions of 
the semicircle D N M, ox 

= T V- times the semicircle D N M \- h). 
h ^ g \2 / 

Hence we have the time required to describe the arc A M- 

2 li g g 

and since the time for ascending the arc if ^ is equal to it, we have 
for the time required to describe the whole arc A M B 

f = 2t^:=27T\/- = 7T i/— . 

ff 9 

Since this quantity is entirely independent of the length of the 

arc, it follows that the times of the oscillations for all arcs of the 
same cycloid are mathematically exactly equal, or that the cycloidal 
pendulum is perfectly isochronal. If we compare this formula with 
the formula for the duration of the oscillations of a circular pen- 
dulum, we find that the durations are the same for both pendulums, 
when the length of the circular pendulum is four times the radius 
of the generating circle of the cycloid. 

Remabk. — In order to make a body suspended by a flexible cord oscil- 
late in a cycloid and thereby to form a cycloidal pendulum, we must hang 

the same between two arcs C and C Oj, 
Fig. 538, of a cycloid, so that during 
each oscillation the cord will unwind 
from one and wind upon the other arc. 
It can easily be shown that, when the 
cord G P wraps and unwraps, the end 
P describes a cycloid equal to the given 
one, but in an inverted position. The 
length of the semi-cycloid is C A =: 
CD = 2A B and the arc ^ is = the 
straight line P, which has been un- 
wound ; but the arc A = twice the 
chord A P = 2 G 0, and therefore 



Fig. 538. 




326.] 



THE ACTION OF GRAVITY, ETC. 



659 



P O = G = A F and H N = A E. Describing upon D H = A B 2^ 
semicircle D K H and drawing the ordinate N P^ we have K H = P G 
-and, therefore, also 
PK = G H = AH —A G = AR-FO = aic AFB— arc AF = 

arc B F = arc B K, 
and, finally, iVP is =: the ordinate iV jK" of the circle plus the correspond- 
ing arc B K', JSfPis therefore the ordinate of a cycloid B P A corre- 
sponding to the generating circle B K H. 

Upon the application of cycloidal pendulums to clocks, see " Jahrbii- 
cher des polytechn. Institutes in Wien," Vol. 20, Art. II. Also Prechtl's 
technologische Encyclopadie, Bd. 19. 

(§ 326.) The Curve of Quickest Descent. — It can be proved 
by the Calculus that the cycloid, besides the property of isoclironism 
or tautochronism, possesses also that of hracliystonism, i.e. it is the 
line in which a body descends from one given point to another in 
the shortest time. 

We can prove this (as Jacob Bernoulli did) in the following 
manner. 

Let the relative position of two points A and B, Fig. 539, be 
given by the vertical distance A C = a and the horizontal one 

B C ^:^ h, and that of a horizontal 
line D E hj the vertical distance 
A D — h; required the point K, in 
which a body falling from A to B 
must intersect the line D E in order 
to reach i? in the shortest time. If 
the body arrives at A with the ve- 
locity V, the velocity at K is 

and supposing that A, K and B are 
infinitely near each other, or that «, l and li are very small com- 
pared to V, we can assume that ^ ^ is described uniformly with 
the velocity v and K B uniformly with the velocity Vx, or that the 
time, in which A K B is described, is 

, AK KB 




V 



Vi 



Denoting D IChj z, we have 



A K= Vh' + z' audi KB = V(a - h)' + {b - z)\ 
and therefore 



660 



GENERAL PRINCIPLES OF MECHANICS 



[§ 326. 



. _ Vh' + z' \/{a - hy + {b - zf 



V 



Vi 



This quantity will be a minimum, when we make its first dif- 
ferential coefficient 

d t z h — z 



But 



d z 'i^Vk'+z' V, V(a - hy + (& - zy 



= 0. 



z 



and 



v¥ + 



KB 
K A 



= COS. A K D ~ COS. 



b -z 



B L 



= COS. KB L ■= COS. 01, 



V{a-hy + {h- zy B K 

if) and </)i denoting the inclination of the paths A K and K Bio the 
horizon ; hence we have for the equation of condition 

COS. <p COS. 01 

V Vi 

Putting the heights due to the velocities v and v^, M A — y and 
NK=:y,,oj: 

V = V2>g y and Vi = V2 g yi, 
our equation becomes 

COS. _ COS. 01 , 

and if we apply this formula to the case of a curved line 8 A K B^ 

COS. (t) 

it follows that for every point of this curve the quotient — ^ must 



Vy 



be a constant quantity, such as 



V2 



This property corresponds to a cycloid 8 G M, Yig. 540 ; for 
we have for an element G K oi this curve 



Fig. 540. 



8 






1 


=: 




A\ '^ 


r^ /_ 






/ 
/ 




,c \ ) 


/ 




\. 




i^rf 


/ 

H // ^ 


y^ 




iK^ 







M 



§337.] THE ACTION OF GRAVITY, ETC. 661 

COS, <t> - Q j^- p2d ~ VM H,EM ~ ^ E M ' ^ 2 r' 
and therefore 



COS. (f) 



Vy V2r 

r denoting the radius C M = C E oi the generating circle E F M, 
Afi arc S G of a cycloid is therefore the arc in luhich a tody 
descends in the shortest time from one jpoint S to another point G. 

§ 327. The Compound or Material Pendulum. — In order 
to determine the duration of an oscillation of a compound pendulum 
or of any body A B, Fig. 541, oscillating about a horizontal axis G, 
we must first find the centre of oscillation (Fr. centre 
^^* ' d'oscillation, Ger. Mittelpunkt des Schwunges or 
J^^ Schwingungspunkt), i.e., that point K of the body 
which, if it oscillates alone around C or forms a 
simple pendulum, has the same duration of oscilla- 
tion as the entire body. We can easily perceive 
that there are several such points in a body, but we 
generally understand by it only that one, which 
lies in the same perpendicular to the horizontal 
axis as the centre of gravity does. 

From the variable angle of displacement K C F ^^ we obtain 
the acceleration of the isolated point K, which is 

= g sin. (^ ; 
for we can imagine that it slides down a plane, whose inclination is 
KHR = KGF=(p. If Jf F is the moment of inertia of the 
entire body or system of bodies A B, M s its statical moment, i.e. 
the product of the mass and the distance G S = s of its centre of 
gravity from the axis of oscillation G, and r the distance G K of 
the centre of oscillation from the axis of rotation or the length of 
the simple pendulum, which vibrates isochronally with the material 
pendulum A B, we have the mass reduced to K 

_M¥ 

- r' ' 
and therefore the rotary force reduced to this point is 

= - M g sin. </>; 

consequently the acceleration is 



662 GENERAL PRINCIPLES OF MECHANICS. [§ 327. 

force s ,^ . , MJc" M s r 

^ mass r ^ ^ ^2 Mk ^ ^ 

In order that the duration of an oscillation of this pendulum 

shall be the same as that of the simple pendulum, it must have in 

eyery position the same acceleration as the other ; hence 

Ms r 

j^^ . g sm. (p = g sm. 0. 

This equation gives 

M h^ __ moment of inertia 
Ms statical moment 

We find, then, that the distance of the centre of oscillation from 
the point alout which the rotation takes ^j'/o^ce, or the length of the 
simple pendulum having the sam,e duration of oscillation as the com- 
pound pendulum, is equal to the moment of inertia of the compound 
pendulum divided hy its statical moment or the moment of its weight. 

Substituting this value of r in the formula zf = tt i/ — , we ob-. 
tain for the duration of an oscillation of a compound pendulum 

t = n y = ni/ ■ — , 

^ Mg s ^ gs' 

or more accurately 

\ Sr/ ' g s 
By inversion we obtain from the duration of an oscillation of a 
suspended body its moment of inertia by putting 

Mk'^= (^-^\ Mg sotF = (^-^^ gs. ' 

Remark — 1) In order to determine the moment of inertia If P of a 
body from the duration of one of its oscillations, it is necessary to know its 
statical moment M g s — G s. The latter is found by drawing the body 
A C, Fig. 542, out of its position of equilibrium by means of a rope J. ^D, 
which passes over a pulley and to which a weight P is suspended. The 
perpendicular C i\r, let fall from the axis C upon the direction of the rope 

A B, is the arm a of the weight P, and P a is equal to the moment G . C H 
of the weight G, which acts vertically at the centre of gravity 8. Denoting 
by a the angle V 8 =: G 8 H, which the body is raised by the weight P, 
we have 

G H = G 8 sin. a = s sin. o, 
and therefore 

G s sin. a = P a, 

from which we deduce the required statical moment 

G s = -7 — . 
stn. a 



§ 327.] 



THE ACTION OF GRAVITY, ETC. 



663 



2) A very simple and useful pendulum A D F^ Fig. 543, may be made 
of a ball of lead A about 1 inch in diameter, suspended by a silk thread, 

Fig. 542. Fig. 548. 

G 





whose up23er end is fastened into a ferrule Z> by a clamping screw. This 
ferrule has upon its end a screw, which passes through- the arm E F 
and is made fast by a nut O^ when the arm has been screwed into a 
door-frame or some other solid support. If the length is C ^ = 0,2485 
or nearly i meter, then this pendulum will beat half-seconds for almost 
an hour, although the arcs in which it oscillates will continually decrease. 
Example — 1) If the point of suspension of a prismatical rod A B, 
Fig. 544, is at a distance G A = l^ from one end A and C B =: l^ 
from the other 5, its moment of inertia, when F denotes its cross-sec- 
tion, is (§ 286) 

M¥ = ^F{1^' -+- Zg'), 
and its statical moment is 

Ms = iF{l^'-l,'); 

fience the length of the simple pendulum, which oscillates isochroually, 
is 



Fig. 544. 
B 

[H 



r = 



M k"" 
Ms 



— 2 



•7 2 _ 7 2 



6^ "' 



11 



I denoting the sum l^ -\- l^ and d the difference Zj — l^, 
rod should beat half-seconds, we must make 



If this 



9 






1 . 39,15 = 9,79 inches, 
and if the rod is 12 inches long we must put 



144 4- 3 cP 
9,79 = , Z_ or t?3 _ 19,58 ^ = _ 48, 



hence 



d = 



Qd 
19.58 - V 191,3764 



19,58 - 13,83 



= 2| inches ; 



from which we obtam 



664 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 328. 






= 6 - ItV = 4^. 




r 



2) If G^ is the weight and I the length of the rod of a pendulum with 
a spheroidal bob A B, Fig, 545, and if K is the weight and r^ the diam- 
eter MA = M B of the latter, we will have 

If the wire weighs 0,05 pounds and the ball 1,5 pounds, and 
if the length of the wire is 1 foot and the radius of the ball 1,15 
inches, we have the distance of the centre of oscillation of this 
pendulum from the axis of rotation 

i . 0,05 . 13^ + ^(13,15=^ + I . 1,15=^) 3,4 + 260,177 



r = 



i . 0,05 

362,577 
30,03y 



13 + 1,5 . 13,15 
= 13,113 inches. 



0,3 + 19,735 



260 177 
If we neglect the wire, r = ' = 13,190 inches, and if we assume 



The 



19,725 
the mass of the ball to be concentrated at its centre r = 13,15 inches 
duration of an oscillation of this pendulum is 

/^ /13 112 

e = TT y - = 0,554 y — ^- = 0,554 yi,0936 = 0,5791 seconds. 

§ 328. Reciprocity of the Point of Suspension and the 
Centre of Oscillation. — The point of suspension and the centre 

of oscillation are reciprocal (Fr. reciproqne ; Ger. wechselseitig), 
I.E. one can be changed for the other, or the pendulum can be sus- 
pended at the centre of oscillation without changing the duration 
of the oscillation. This can be proved, by the aid of what was 
said in § 284, in the following manner. Let W be the moment of 
inertia of the compound pendulum A B, Fig. 546, referred to an 
axis of rotation passing through its centre of grav- 
ity S, for an axis of rotation passing through C, 
which is at a distance C S = s from the centre of 
gravity 8, we have 

W.^W^- Ms\ 
and therefore the distance of the centre of oscilla- 
tion from the axis of rotation C is 

_W,_ W -}- M s' _ W 
^ ~ Ms ~ WJ^^ ~ Ms 
Denoting the distance X S — r — s of the centre 
of oscillation X from the centre of gravity by Si, we obtain the 

equation s Si = -ji^, in which s and Si present themselves in the 




+ s. 



§ 329.] 



THE ACTION OF GRAVITY, ETC. 



665 



iQ 



same manner, and therefore can be changed for one another. This 
formula is consequently applicable not only to the case, where s 
expresses the distance of centre of rotation and Si that of the cen- 
tre of oscillation from the centre of gravity, but also to the case, 
where s expresses the distance of the centre of oscillation and Si 
that of the centre of rotation from the centre of gravity. There- 
fore becomes the centre of oscillation, when K becomes the point 
of suspension. We employ this property in the revei^s- 
Fig. 547. ^j/g pendulum A B, Fig. 547, first suggested by Bohnen- 
^ berger and afterwards employed by Kater. It is provided 
with two knife-edge axes C and K, which are so placed, 
that the duration of an oscillation remains the same, 
whether the pendulum is suspended from one axis or the 
other. In order to avoid changing the position of the 
axes in reference to each other, two sliding weights are 
applied to it, the smaller of which can be moved by a 
small screw. If by sliding the weights we have brought 
them to such a position, that the duration of an oscilla- 
tion is the same, whether the pendulum be suspended in 
C or K, we obtain in the distance C K the length r 
of the simple pendulum, which vibrates isochronally with 
the re ver sable pendulum, and the duration of the oscilla- 
tion is given by the formula 

t = n -/^. 

§ 329. Rocking Pendulum. — The roching of a body with a 
cylindrical base can be compared to the oscillation of a pendulum. 
This rocking, like every other rolling motion, is composed of a mo- 
tion of translation and one of rotation, but we can consider it as a 
rotation about a variable axis. This axis of rotation is the point 
of support, where the rocking body ABC, Fig. 548, rests upon 

the horizontal support HE. Let 
the radius CD— C F of the cylin- 
drical base A D B he = r and the 
distance C S of the centre of gravity 
S of the whole body from the centre 
C of this base be = s, then we have 
for the distance 8 P — y of the cen- 
tre of gravity from the centre of rota- 
tion, corresponding to the angle 

S C P =: (P, 




666 GENERAL PRINCIPLES OF MECHANICS. [§ 329. 

if — T' + 5"" — 2 r 5 C08, (^ — {r — sf + 4 r s (sin, -^ I- 

If we denote the moment of inertia of the whole body in reference 
to the centre of gravity /S' by if F, we obtain the moment of inertia 
in reference to the point of support P 

W = M{¥ -\- f) = if [f + (r-sY ■^4:rs (^sin, ^\\ 

for which for small angles we can put M [h"^ + (r — sY -{- r s (f)''] or 
even if [F + (r — sY]- Now since the moment of the force = 
G.SN=Mg.CS sin. (p = M g s sin. (f), we have the angular 
acceleration for a rotation around P 

__ moment of force _ M g s sin. _ 9 ^ ^^'^' 



moment of inertia M\]c^ + (r — sY'\ h"^ + {r — sY' 
For the simple pendulum it is = ~ — ^, when Ti denotes its length. 

If they should oscillate isochronally, we must have 

g s sin. (p _ g sin. (p , ^^^'^ ^ (^' '~ ^)^ 

T"2 \ 1 \i — ? I.E.J T\ — . 

F 4- (r — sY Tx s 

The duration of an oscillation of the rocking hody is, therefore, 

g 9s 

This theory is applicable to a pendulum A B, Fig. 549, 
with a rounded axis of rotation C M, when we substitute 
for r the radius of curvature C if of this axis. If instead 
of the rounded axis a knife-edge axis D is used, the dura- 
tion of an oscillation would be 





a/¥ +D S' ./¥ + ( s - xY 

g .P S ^ g[s - X) 

when the distance C P of the knife-edge P from the cen- 
tre C of the rounded axis is denoted by x. The two pen- 
dulums will have the same duration of oscillation, when 

F + {s-xY F 4- (r - sY F F 4- r' _ 
^^ ^ = -^ ^, or — X = — 2 r; 

S — X s S — X s 

Tc^ F 1^ X 

putting approximatively — 1 -,- and neglecting r', we 

S X s s 

obtain _ 2 r s'* 

^ ~ 5^^^F* 

Remark. — The conical pendulum will be discussed in the third part, 
in the article upon the '' Governor." 

In the appendix to this volume the subject of oscillation is treated at 
length. 



^. 330.J 



THE THEORY OF IMPACT. 



^67 



CHAPTER IV. 



THE THEORY OF IMPACT. 



§ 330. Impact in G-eneral. — On account of the impenetra- 
bility of matter, two bodies cannot occupy the same space at the 
same time. If two bodies come together in such a Avay that one 
seeks to force itself into the space occupied by the other, a recipro- 
cal action between them takes place, which causes a change in the 
conditions of motion of these bodies. This reciprocal action is 
what is called impact or collision (Fr. choc, Ger. Stoss). 

The conditions of impact depend, in the first place, upon the 
law of the equality of action and reaction (§ 65); during the im- 
pact one body presses exactly as much upon the other as the other 
does upon it in the opposite direction. The straight line, normal 
to the surfaces, in which the two bodies touch each other, and 
passing through the point of tangency, is the direction of the 
force of impact. If the centre of gravity of the two bodies is upon 
this line, the impact is said to be central; if not, it is said to be 
eccentric. When the bodies A and B, Fig. 550, collide, the impact 

Fig. 550. 





is central ; for their centres of gravity Si and S^ lie in the normal 
iViVto the tangent plane. In the case represented in Fig. 551 the 
impact of A is central and that of B eccentric ; for S^ lies in and 
Si without the normal line or line of impact JV M 

When we consider the direction of motion, we distinguish direct 
impact (Fr. choc direct, Ger. gerader Stoss) and oblique impact (Fr. 
choc oblique, Ger. shiefer Stoss). In direct impact the line of im- 



MS 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 331. 



Fm. 552. 



^y a 



pact coincides with the direction of motion ; in obHque impact the 
two directions diverge from each other. If the two bodies A and 

By Fig. 552, move in the directions 
Si Ci and S^ C^, which diverge from 
the line of impact N N, the impact 
which takes place is oblique, while, 
on the contrary, it would have been 
direct if the directions of motion had 



coincided with N N. 

We distinguish, also, the impact' 
of free bodies from that of those par- 




tially or entirely retained. 



§ 331. The time during which motion is imparted to a body or 
a change in its motion is produced is, it is true, very small, but by 
no means infinitely so ; it depends not only upon the force of im- 
pact, but also upon the mass, velocity and elasticity of the colliding 
bodies. We can assume this time to consist of two parts. In the 
first period the bodies compress each other, and in the second they 
expand again, either totally or partially. The elasticity of the 
body, which is brought into action by the compression, puts itself 
into equilibrium with the inertia, and thus changes the condition 
of motion of the body. If during the compression the limit of 
elasticity is not surpassed, the body returns to exactly its former 
shape, and it is said to be perfectly elastic ; but if the body, after 
the impact, only partially resumes its original form, we say it is 
imperfectly elastic ; and if, finally, the body retains the shape it as- 
sumed under the maximum of compression or possesses no ten- 
dency to re-expand, we say that the body is inelastic. This classi- 
fication of impact is correct within certain limits only ; for it is 
possible that the same body will act as an elastic one when the im- 
pact is slight, and as an inelastic one when the impact is vioicnt. 
Strictly speaking, perfectly elastic and perfectly inelastic bodies 
have no existence ; but we will hereafter consider elastic bodies to 
be those which apparently resume their original form, and inelastic 
bodies to be those which undergo a considerable change of form in 
consequence of the impact. 

In practical mechanics the bodies, such as wood, iron, etc., 
which are subjected to impact, are very often regarded as inelastic, 
because they either possess but little elasticity or lose the greater 
part of their elasticity in consequence of the repetition of the im- 



332.] 



THE THEORY OF IMPACT. 



669 



pact. It is very important in constructing machinery, etc., to avoid 
impacts as much as possible. If this cannot be done, we should 
diminish their intensity or change them into elastic ones ; for they 
give rise to jars or concussions and cause the machinery to wear 
very fast, and in consequence a portion of the energy of the ma- 
chine is consumed. 



§ 332. Central Impact.— Let us first investigate the laws of 
the direct central impact of bodies moving freely. Let us suppose 
the duration of the impact composed of the equal elements r, and 
the pressure between the bodies during the first element of time to 
be = Pi, during the second to be = P^, during the third to be 

= Pg, etc. Now if the mass of the 
body A, Fig. 553, = Mi, we have the 
corresponding accelerations 
P, P, 




- y ^ ^m 



J?i - -^^ V^ - j^^ 



Pz = -^^ etc. 

But, according to § 19, the vari- 
ation in velocity corresponding to p 
and to an element of the time r is 



K 



pt; 



hence the elementary increments and diminutions of velocity in 
the foregoing case are 






P3T . 
K^ — -j^, etc., 



and the increase or decrease in velocity of the mass M^ after a cer- 
tain time is . 

«1 + «2 4- «3 + . . . = (Pi + P2 + P3 + . . .) ^> 

and the corresponding variation in velocity of the body B, whose 
mass is Mc^, is 

= (P, + P2 + P3 + . . .) ~. 

The pressure acts in the following or impinging body in oppo- 
sition to the velocity c, producing a diminution of velocity, and 
after a certain time the velocity, which the body still possesses, is 



670 GENERAL PRINCIPLES OF MECHANICS. [§333. 

^1 = Ci — (A + Ps + . . .) ^. J^ 

The pressure acts upon the body B, which is in advance and which 
is impinged upon, in the direction of motion, its velocity d is 
increased and becomes 

t;^ = ^2 + (Pi + Ps + A + . . .) ^. 

Ehminating from the two equations (Pi + Pa + P3 4- . . .) r, 
we have the general formula 

I. ifi (ci — Vi) = Jfs (^2 — C2), or 

M^ Vi + M^V^ — i/i Ci + ilfs Ca. 

The product of the mass of a body and its velocity is called its 
momentum (Fr. quantite de mouvement ; Ger. Bewegungsmoment), 
and we can consequently assert that at every instant of the impact 
the sum of the momentums {M^ v^ + M^ v^) of the two bodies is the 
same as before the impact tooh place. 

At the instant of greatest compression, the two bodies have the 
same velocity v, hence if we substitute this value v for Vx and ^'2 in 
the formula just found, we obtain 

M,v ^ M,v = M,c, + M., c„ 

from which we deduce the velocity of the bodies at the moment of 
greatest compression 

_ i fi gi + i/2 ga 

"" - ~ if 1 + Jf 2 • 

If the bodies A and B are inelastic, i.e. if after compression 
they have no tendency to expand, all imparting or changing of 
motion ceases, when the bodies have been subjected to the maxi- 
mum compression, and they then move on with the common 
velocity 

Jfi Cx + ifs C2 



V 



ifi + M, 



Example — ^1) If an inelastic body B weighing 80 pounds is moving 
with a velocity of 3 feet and is impinged upon by another inelastic body 
A weighing 50 pounds and moving with a velocity of 7 feet, the two move 
on after the collision with a velocity 

_ 50 . 7 + 30 . 3 _ 350 + 90 _ 44 _ 11 _ 
"^ - "'5F+'30 - 80 " - 8 - Y - ^^ ^^^*' 

2) lo order to cause a body weighing 120 pounds to change its velocity 



§333.] THE THEORY OF IMPACT. 671 

from c = 1^ feet to « = 2 feet, we let a body weighing 50 pounds strike 
• it ; what velocity must the latter have ? Here we have 

., = . + (^LZL^l = . + (A:^^i?5 = . 4 = B,M-eet. 

§ 333. Elastic Impact. — If the colliding bodies are perfectly/ 
elastic, they expand gradually during the secdnd period of the im- 
pact after having been compressed in the first one, and when they 
have finally assumed their original form, they continue their mo- 
tion with different velocities. Since the work done in compressing 
an elastic body is equal to the energy restored by the body, when 
it expands again, no loss of vis viva is caused by the impact of 
elastic bodies. Hence we have for the vis viva the following equa- 
tion 

11. Jfi V,' + M, v^ r= M^ c^ + 3U c,\ or 

From equations I. and II. the velocities v-^ and v<^ of the bodies 
after the impact can be found. First by division we bave 

g i' — Vi _ v j — c^ 

I.E., 

^1 + ^1 = ^2 + Cs, or v^ — Vx — c-^ — c<i\ 
substituting the value 

^^2 = Ci + ^1 — Cs, 

deduced from the last equation, in equation I., we have 

J!/; Vi + ilSfs t^i + M^ (c, — c^ = M,c, + M, c„ or 

whence 

Hence if the bodies are inelastic, the loss of velocity of one 
body is 

r -v = r - ^' ^' '^ ^' ^' - ^' ^^' ~ ^^) 
and when they are elastic, it is double that amount, or 



672 GENERAL PRINCIPLES OF MECHANICS. [§334. 

''^ ~ ""^ - Jf, + Jf, ' 

and while for inelastic bodies we have the gain in yelocity of the 
other body 

_ — ^1 Oi + M c, _ M, (c i — c) 

"" '''- M, + M, ""'^ M, + M,' 

for elastic bodies it is 

_ _ 2 Mi (gi - c) 

""' '''-' M^ + M, ' 
or double as much. 

Example. — Two perfectly elastic balls, one weighing 10 pounds and 
the other 16 pounds, collide with the velocities 12 and 6 feet. What are 
their velocities after the impact ? Here M^ = 10, c^ = 12, ifg = 16 and 
Cg = — 6 feet, and the loss of velocity of the first body is 

3. 16 (12 + 6) 2. 16 . 18 ^„,^, , 
«i - ^1 = 10 + 16 = 26 = ^^'^^^ ^"^^** 

and the increase of the velocity of the other is ' 

2. 10 . 18 ,,^,,, , 

^^2 - ^2 = K7i =13,846 feet. 

2 2 26 ' 

The first body, therefore, rebounds after the collision with the velocity 

-Wj = 12-22,154 = — 10,154 feet, and the other with the velocity «)g = — 6 + 

13,846 = 7,846 feet. The vis viva of these bodies after the impact is 

= ifi v^' + M^ v^"" = 10 . 10,154=^ + 16 . 7,846^ = 1031 + 985 = 2016 or 

the same as that before impact ifj c^ + ifg Cg = 10 . 12^ + 16 . 6^^ = 1440 + 

576 = 2016. 

If the bodies were inelastic, the first body would lose but -^— ^r — - 

2 

= 11,077 feet of its velocity and the other would gain ^^ ~ ^^ = 6,928 

feet; the velocity of the first body after the impact would be 12 — 11,077 = 
0,923 feet, and that of the second — 6 + 6,923 = 0,923 ; a loss of me- 
chanical effect 

[2016 - (10 + 16) 0,923^^] :2g = (2016 - 22,2) . 0,0155 = 30,9 foot-pounds, 
however, takes place. 

§ 334. Particular Cases. — The formulas found in the fore- 
going paragraph for the final velocities of impact are of course 
applicable, when one of the bodies is at rest, or when the two 
bodies move in opposite directions and towards each other, or 
when the mass of one of the bodies is infinitely great compared 
to that of the other, etc. If the mass M^ is at rest, we have c^ — 
and therefore for inelastic bodies 



§ 334.] THE THEORY OF IMPACT. 673 

and for elastic ones 

v, = c,- Z- "?' ^Ir ~ 1j , "i^' ^i> and 

t^2 = + 



V - 


" ifi + Mi 




2 M, 


Ci 


M,- 


ilf^ 


M, + 


M, ~ 


M, H- 


M, 


2 if, 


Ci 


2i1/ 


1 



Jtfl + Jf 2 i^i + M, ' 

If the bodies move towards each other, c^ is negative, and there- 
fore for inelastic bodies 

i; = — r^ r— — , and tor elastic ones 

if 1 + i/a 

If in this case the momenta of the bodies are equal, or M^ Ci = 
i/g Cci, when the bodies are inelastic, ^^ = 0, i.e., the bodies bring 
each other to rest, but if they are elastic, 

Vi = Ci m'^m ' = Ci- 2ci --= - Ci, and 

2 (if^ c, + i/i g^) _ , 9 . _ , 

the bodies after the impact proceed in the opposite direction with 
the same velocity they originally had. If, on the contrary, the 
masses are equal, we have for inelastic bodies 

Ci c^ 

" = -%-' 

^d for elastic ones 

Vx— — Cci and v^ = Ci, 

I.E., each body returns with the same velocity that the other body 
had before the impact. If the bodies move in the same direction, 
and if the one in advance is infinitely great, we have for inelastic 
bodies 

_ if 2^2 _ 

and for elastic ones 

t^i = Ci — 2 (ci — C2) = 2 C2 — c„ ?;2 = Ca + = C2 ; 
the velocity of the infinitely great body is not changed by the 
impact. If the infinitely great body is at rest, or if c^ = 0, we have 
for inelastic bodies 

V = 0, 
and for elastic ones 

Vi = — Ci,Vi = 0; 
here the infinitely great body remains at rest; but in the first case 



G74 GENERAL PRINCIPLES OF MECHANICS. [§335. 

the impinging body loses its velocity completely, and in the second 
case it is transformed into an equal opposite one. 

Example — 1) With what velocity must a body weighing 8 pounds 
strike a body weighing 35 pounds in order to communicate to the latter a 
velocity of 3 feet ? If the bodies are inelastic, we must put 

1) — -— — ^ \ , I.E. 2 = * 



ifi + iYg' • • 8 + 35' 

whence we obtain c^ = ^-^ = 8|- feet, which is the required velocity ; if 
they were elastic, we would have 

®2 = -^TF — ^ .V ^ whence c. = ^ = 44 feet. 

3) If a ball if i, Fig. 554, strikes with the velocity c^ the mass M^ =n ifj, 

Fig. 554 which is at rest, if the second mass 

jyj strikes a third if 3 = n ifg = n^ M^, 

^ ' " ^'^ with the velocity imparted to it by 



the impact, and if this third mass 
4 strikes a fourth M^ = n M^ = 
n^ M^, etc., we have, when these 
masses are perfectly elastic, the velocities 

_ 2ifi _ __3_ _ 3ifg _ 3 _ 




(iT^y^-^^ = (r!-^^- 



If, for example, the weight of each mass is one-half that of the pre- 
ceding one, we have the ratio of the geometrical series formed by the 
masses 



n = h 



hence 

«3 =f<'i,'»3 =(f)'«i,«4 =(t)'^i...,«io = (|)'^i = 13,33.^1. 

§ 335. Loss of Energy. — When two inelastic bodies collide, 
a loss of vis viva always takes place, and therefore they do not 
possess so much energy after the impact as before. Before the im- 
pact the vis viva of the masses Mi and M^, which move with the 
velocities Ci and d, is 

M, c' + M, c,\ 

but after the impact they move with the velocity 

V = — W i^r — and 

M, -f i/2 



§335.] THE THEORY OF IMPACT. 675 

their vis viva is 

by subtraction we obtain the loss of vis viva caused by the impact 

X = M, {c' - v') + M, (ci - v') 

= 3fr (ci + v) ((?i — v) — 31^ (^2 + v) {v — c), but 



M, {c, - v) - M, {v - c,) = V, + b^— ^ 



whence 



If the weights of the bodies are G^ and 6^2? or if 

M, = ^ and if2 = -, 
9 9 

we have the loss of energy or the work done 

A _ (gl — C^Y ^1 ^ 2 

2^ ''g;^g: 

C C 

We call -~ — ~ the liarmonic mean between G^^ and 6^2» and we 

can assert that the loss of energy, caused hy the impact of tivo inelastic 
bodies and expended in changing their form, is equal to the product 
of harmonic mean of the tiuo masses and the height due to the differ- 
ence of their velocities. 

If one of the masses 31^ is at rest, we have the loss of 
mechanical effect 



A = 



Cx G^ G^ 



% g' G, + G,' - 

and if the moving mass M^ is very great, compared to the mass at 
rest, G^ disappears before G^ and the formula becomes 

2g 
We can also put 
K= 3f, {c' - v') + M, (c-^ - v') 

= 3f,(c,'-2c,v + v' + 2c,v-2v')-h3f,{c,''-2c,v + v'-h2c,v-2v,') 
= 3f, (c, -vY +2 31, V (c, -v) + M, {c, - vY -h 2 3f, v {c, - v) 
=. i¥i (c, - vY + 3f, {c, - vY ; 
for 3f, {c, - v) = M,{v - cs). 

Prom this we see that the vis viva lost by the inelastic impact is 



676 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 336. 



equal to the sum of the products of the masses and the squares of 
their gain or loss of velocity. 

Example — 1) If in a macliine 16 impacts per minute take place be- 
tween the masses 



M, 



1000 



lbs. and M^ 



1200 



lbs., 



9 9 

whose velocities are c^ = 5 feet and Cg = 3 feet, the loss of energy, in con- 
sequence of these impacts, is 

(5 — 3)2 1000 . 1200 



^ = i^ 



2200 



tV . 9 . 0,0155 ^^ = 20,29 foot-lbs. 



^ • % g ' 
per second. 

2) If two trains of cars, weighing 120000 and 160000 pounds, come into 
collision upon a railroad when their velocities are Cj = 20 and Cg = 15 
feet, a loss of mechanical effect, which is expended in destroying the loco- 
motives and cars, ensues ; its value is 



^20 + 15y 120000 ■ 1 60000 _ ^^^ 1920000 

2 g I' 280000 " ^^ ' "'^^^^ ' ~28 



_ / 20 + i5 y 



= 1302000 foot-lbs. 



§ 336. Hardness. — If we know the modulus of elasticity of 
the colliding bodies, we can find also the compressive force and the 
amount of compression. Let the cross-section of the bodies A and 

B, Fig. 555, be F^ and F^, their length 
li and 4j and their moduli of elasti- 
city be F^ and F^. If they impinge 
upon one another, the compressions 
produced are, according to § 204, 

Ai = -- — ]= and Ag = 



-K 







Fi Fi 
and their ratio is 
Fi F<i li 
Fx Fx 4 



jFo ^2 



If, for the sake of simplicity, we denote -^ — ^ by Hx and ^ "^ 



by -STs, we obtain 



and 



k 



P P 

fix ^^ 'xf and Ag =: -^1 
Mx Jdi 



U 






El 



Calling, with Whewell (see the Mechanics of Engineering, 

F F 
207), the quantity ~z— the hardness (Fr. durete raideur, Ger» 



I 336.] THE THEORY OF IMPACT. 677 

Harte) of a body, it follows that the depth of compression is in- 
versely proportional to the hardness. 

C 

If the mass M = — impinges with the velocity c upon an im- 

movable or infinitely great mass, all its vis viva is expended in com- 
pressing the latter body, whence, according to § 206, 

But the space o is equal to the sum of the compressions /t^ and 

P P 

Ag, and we have X^ = — and A.^ == ^, whence 

or inversely „ _ H^ H<i 

Substituting this value of P in the above equation, we obtain the 
equation of condition 

^ • ^, + ^/ ~ 2 ^ ^' 
or JH, ^ H, a 



= c\/- 



HiH, ' g' 
by the aid of which the values P, Aj and X^ can be calculated. 

Example.— If with a sledge, that weighs 50 pounds and is 6 inches long 
and the area of whose face is 4 square inches, we strike a lead plate one 
inch thick, and the area of whose cross-section is 2 square inches, with a 
velocity of 50 feet, the effect can be discussed as follows. Assuming ^^ = 
29000000 as the modulus of elasticity of iron and E^ = 700000 as that of 
lead, we find the hardness of the two bodies to be 

. „ F.E. 4.29000000 ,^„„„^„„ , 
E^ = ^=— i = = 19333333 and 

F^E„ 2.700000 ,,^r.r.r.r. 

E^ = --V^ = z = 1400000. 

tg 1 

Substituting these values in the formula 



(J = c y ■ 



E^ + E^ G 



and putting the weight of the sledge = 4.6, 0,29 = 7 pounds, or 

G 

— = 7 . 0,031 =: 0,217, 

we have for the space described by the sledge in compressing the lead 

_ /■~20733333To;217~ ^^ /0,44991 
'-^^V 19333333. 1400000 = ^^ r 2706666=^'^^^^ inches=0,245 lines. 



678 GENERAL PRINCIPLES OF MECHANICS. [§337. 

Hence the pressure is 

H^ H„ 19333333 . 1400000 ^ , 

^ = stVW, ' ' = 207333 31 ' ^'^^^^ = ^^^^^ P^^^^« 5 

the compression of the hammer is 

P 36632 . . . , 

^1 = ^ = f 933333 3 = 0,0014 inches = 0,016 lines, 

and that of the lead 

P 26632 

^g = ^ = 1400000 ^ ^'^^^ ^^^^^® ^ 0,228 lines. 

§ 337- Elastic -inelastic Impact. — If two masses i/j and M^ 
are moving with the velocities Cx and c^ in the same direction, their 
common velocity at the moment of maximum compression is, ac- 
cording to § 332, 

"^ ~ M, + M^~' 
and the work done during the compression, according to § 335, is 

. ^ (gi - c,y M, M, ^ (c, - c,y G, G, 

2 ' M, -h M, 2g ' G, + G,' ^ 

but this mechanical effect can be put 

= 1 P <7 = -1 P (A, + A,) = ^ gf^ "". 

whence we obtain for the sum of the compressions of the two 

masses 



_ / _ ^ \ a/ Gi Gc^ Hx + H j 

from which the compressive force P and the compressions Aj and 
As of the two masses can be found. 

If the bodies are inelastic, they remain compressed after the 
impact ; but if one only is inelastic, the other resumes its original 
form in a second period, and the work done in expanding produces 
another change of velocity. If, for example, the mass M^ = 

— is elastic, the work done in the second period of the impact is 

^ (^1 - c,Y Gi G2 H, 

2g ' G, + G,' H, + H,' 

We have, therefore, when the velocities after the impact are v^ and 
v<i, the formulas 



§337.] THE THEORY OF IMPACT 679 

Ml Vi + M^Vi=. Ml Ci + J/a C2 and 

MM H 

M, Vr' + M, v^ = Ml c;' + M, c,^ + {ci - c,y . j^^^^^^ • j ^^ ^ jj-^ 

,^ o / V, M M^ , ., M, M, I-L 

I.E. 

MM TT 

Ml vi^ + if, v,^ = Ml ci' + M, c/ - (c, - c,y . j^^-;^^ . ]^;^^; 

If we put the loss of velocity Ci — Vi = x, we have the gain in 

velocity 

_ MiX 

""^ ~ ""^ - ~1^' 
and the last equation assumes the following form : 

\ /o Mix\ ., M, Hi _ 

x{^ci-x)-xyzc, + -j^j - {ci - c) ^^-:^rif, * hVVh, ~ ^' 

or 

j^^^ X x5 {ci c,)x + Ui ^'^) ' M, + M,' Hi-^ H,- ^' 

M 

Multiplying by -^7 — ^jj and remembering that 

Ifli -f- luti 

III ^ _ H. 



Hi + H, Hi + /// 

we obtain the quadratic equation 



« o / ^ ^2 I X2 / M^ \ 

X - ^ (c, - c) j^^^^tm, ^ + ^^^ - ^^) \m, + MJ 

_ . _ ., / M, V ^. 
~ ^""^ ""'^ \Mi + W ^1 + H, 
or 

/ if. V_ , ., / if. \ 

r ~ ^''^ " ""'^ MTVmJ ~ ^""^ ~ ""'^ \Mi + mJ ' Hi 



H, 



by resolving which we obtain the loss of velocity x of the first body 

«.-..= (c, - 0,) ^^^ (1 + Vjjf-^), 

and the gain of velocity of the other 

v.-c. = (0. - oO 3^^^ (1 + -Z^^^.) 

Example. — If we assume that in tlie example of the foregoing para- 
graph the iron sledge is perfectly elastic and that the lead plate is perfectly 
inelastic, we obtain the loss of velocity of the hammer, which weighs 7 
pounds and falls with the velocity of 50 feet, since we must put C2 = and 

Jfg = GO, 



680 GENERAL PRINCIPLES OF MECHANICS. [%ddS. 



I . / ^2 \ L ./ 1400000 \ 

c, -., =c, ^1 + Vh^^hJ = 50 ^1 + y 2o^33333J 

= 50 (1 + 0,26) = 63 feet ; 
hence the velocity of the sledge after the blow is 

i)j = Ci - 63 == 50 — 63 = — 13 feet. 
The velocity of the lead plate, which is retained, of course remains = 0. 

§ 338. Imperfectly Elastic Impact. — If the colliding 
bodies are imperfectly elastic, they expand only partially in the second 
period of the impact and the mechanical effect expended in pro- 
ducing the compression in the first period is not entirely restored 
in the second period. If/tjand Ag again denote the amount of 
compression an.d P the pressure (called also the force of distorsion), 
we have the mechanical effects expended during the compression 
= ^ P Ai and I P Ag, and if during the expansion but the /xth 
part or more generally during the expansion of the first body 
but the //^th and during that of the other but the ii^ih part of the 
mechanical effect is restored, the entire loss of mechanical effect is 
A^\P\{1- II,) Ai + (1 - II,) AJ, 

P P 

or, putting Ai =: — and Ag = -=r, 

The force with which the bodies react in the second period is 
called the force of restitutio7i. 

But according to the foregoing paragraph we have 

hence the required loss of mechanical effect is 

A = (^1 ~ ^^y M,M, H, H., n - fi, 1 - ii\ 

2 ' M, •\- M,' H, -{- H,\ Hi H, J 

_ {ci — gg)' M,M, I _ ^1 H^ + ^2 -^A 
~ 2 ' M, + M,\ H, + H, r 

To find the velocities Vi and Vc^ after the impact, we employ the 
equations 

M, Vy + Jfs ^2 = M^Ci + M, ^2 and 
M, v,^ + M, V,' = M, c' + M, c^ 

_ M,M, (i-fi,)ir,+ {i-fi,)ir, 

which we must combine and resolve. In exactlv the same manner 
as in the last paragraph the loss of velocity of the first body is found 
to be 



338.] THE THEORY OF IMPACT. 681 



, ^^ -(e -c) ^' (l 4- i/ffL^?L±JfL^A 
C, --V, - [Cr c,) j^^^ MA ^^ H. + H, /' 

and the gain in velocity of the hody, which is in advance, 



,^^o.^(c,-c.)-^-^-j^^ \l + |/^^^-^^J. 

These two formulas include also the laws of perfectly elastic 
and of inelastic impact. If we substitute in them ii^ =. ^.,— \^ we 
obtain the formula already found for perfectly elastic bodies, and 
if we assum,e [1^ = 11^^ = 0, we obtain the formulas for inelastic im- 
pact, etc. If both bodies are equally elastic, or j^i = ^i^, we have 
more simply 



and 



v,-c, = (c^ - c) ^^ ^' ^^ (1 + VJ). 



If the mass Mc^ is at rest and infinitely great, it follows that 

Ci — V^ = Ci {1 + V 11), I.E., 

Vx =^ — Ci V jjb, or inversely 

If we cause a mass M^ to fall from a height h upon a rigidly 
supported mass i/s? and if it bounces back to a height hi, we can 
determine the coefficient of imperfect elasticity of the body by the 
formula hi *> 

Newton found in this way for ivory, 

f - iW = If = 0,79, 

for glass 

^ = (i|)= = 0,9375^ = 0,879, 

and for cork, steel and wool 

11 = (1)^ = 0,555^ = 0,309. 

We assume, in this case, that the falling body is a sphere and 
that the body upon which it falls is flat. 

General Morin by causing cannon balls, weighing from 6 to 20 
kilograms, to fall upon masses of clay, wood and cast-iron, which 
were suspended from a spring 'balance or spring dynamometer 
found that for clay and wood \i is nearly — 0, and that, on the 
contrary, for cast-iron it is nearly = 1, i.e. that the impact of 



682 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 339. 



Fig. 556. 



bodies of former substances can be considered as inelastic and that 
of those of the latter substances as perfectly elastic (see A. Morin, 
I^otions fondamentales de Mecanique, Art. 67-70). 

Example. — What will be the velocities of two steel plates after impact, if 
before the impact their velocities were c^ = 10 and c^= —6 feet, and if one 

weighs 30 and the other 40 pounds ? Here we have 

16 8 
c, - ^, = (10 + 6) . f^ (1 + I) = 16 . t . ^ = -^ = 14,22 feet, 

hence the required velocities are 

^^ = c^__ 14^22 .= 10 - 14,22 = - 4,22 feet 
and 

^3 = Cg + 10,66 = - 6 + 10,66 = 4,66 feet. 

§ 339. Oblique Impact. — If the directions of motion 8i Cx 
and /Sg O^ of the two bodies A and B, Fig. 556, diverge from the 

normal A^iV^to the tangent plane, 
an oblique impact takes place. The 
theory of oblique impact can be re- 
ferred to that of direct impact by 
decomposing the velocities 8i Cx = Cx 
and S.i Co = c, into their components 
in the direction of the normal and 
tangent ; the components in the di- 
rection of the normal produce a 
direct impact, and are, therefore, changed exactly as in the case of 
direct impact, while the velocities parallel to the tangent plane 
cause no impact, and, therefore, remain unchanged. If we combine 
the normal velocity of any body, obtained according to the rules 
for direct impact, with the tangential velocity, which has remained 
unchanged, the resultant is the velocity of the body after the im- 
pact. Putting the angles formed by the directions of motion with 
the normal equal to a^ and Cg, or Cx Sx N ^ ax and C^ S^i N=^ a^, we 
obtain for the normal velocities 8x Ex and S^ E.^ the values Cx cos. aj 
and c<i COS. a^, and, on the contrary, for the tangential velocities Sx Ex 
and S2 i^2 the values Cx sin. ax and c^ sin. a^. 

The normal velocities are changed by the collision, the first one 
becoming 



-N- 




Vx = Cx COS. ax — [cx cos. Ci — C2 COS, a^ 
and the second 

V^i = C<i COS. ^2 -I- (Cx COS. «! — C2 COS. tts) 



ifo 



Mx + M, 
Mx 



(1 + ^^), 



(1 + ^^), 



Mx + M, 
in which Jf, and M^ denote the masses of the two bodies. 



§340.] THE THEORY OF IMPACT. 683 

From Vi and c^ sin. a^ we obtain the velocity Si Gx of the first 
body after the impact 

lu^ — Vvi^ -4- ci' sim' a,, 
and from v^ and c.2 sin. a,^ the velocity Sc^ G^ of the second body 

lo.j — Vi\^ + c^^ sin.^ ^2 ; 
the angles formed by the directions of the velocities with the 
normal are given by the formulas 

„ <?i sin. Oi T , n <^2 sin. ac^ 
tang. Pi = — and tang, (j^ = , 

jSj denoting the angle Gi Si N and jSg the angle Gc, S^ iV". 

Example — 1) Two balls, weighing 30 and 50 pounds, strike each other 
with the velocities c^ = 20 and c^ = 25 feet, whose directions form the 
angles a^ = 21° 35' and a^ = 65° 20' with the direction of the normal to 
the tangent plane ; in what direction and with what velocity will these 
bodies move after the impact ? The constant components are 

c^ sin. a^ = 20 . sin. 21° 35' = 7,357 feet and 

Cg sin. ag = 25 . sin. 65° 20' = 22,719 feet, 
and the variable ones are 

c^ COS. a^ = 20 . COS. 21° 35' = 18,598 feet and 

Cg COS. ag = 25 . COS. 65° 20' = 10,433 feet. 

If the bodies are inelastic, we have /« = 0, and therefore the normal 
velocities after the impact are 

«i = 18,598- (18,598 - 10,433) . ff = 18,598 - 5,103 = 13,495 feet and 
ijg = 10,433 + 8,165 . I = 10,433 + 3,062 = 13,495 feet. 
Hence the resulting velocities are 



w^ = Vl3,495^ + 7,357^ = V236,24 = 15,37 feet and 

w^ = Vl3,4952 4- 22,719' = V698^ = 26,42 feet; 

and their directions are determined by the formulas 

7 357 
tang. (3^ = ^^^^, log. tang. 13^ = 0,73653 — 1, i3^ = 28° 36' and 

22,719 , 
tang. (3^ = £3^, hg. tang, p^ = 0,22622, 13^ = 59° 17'. 

§ 340. Impact against an Infinitely G-reat Mass. — If the 

mass A, Fig. 557, strikes against another mass, which is infinitely 
great, or against an immovable object B B, or if Ca = and 
Met = 00 , we have 

Vi = c, COS. ai — Ci COS. a, (1 + Vfi) = — Ci cos. aj VfJi and 

v^ = -h Ci COS. tti —^ ^ =: 4- = 0, 



00 



684 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 340. 



Fig. 557. 



if in addition jit = 0, we have Vi = 0, but if fj, = 1, v^ = — Ci cos. a^, 
I.E., when the impact is inelastic, the normal force is complete- 
ly annihilated, hut, on the contrary, 
when it is perfectly elastic, the normal 
force is changed into an equal opposite 
one. The angle formed by the di- 
rection of motion after the impact 
with the normal is determined by the 
N equation 

Cx sin. «! Ci sin. Q i 




tang. (3i — 



= — tang. 



Ci cos. a^ V^i 



■ «■ ^\ ■ 



for inelastic bodies 



tang, p, = ^^^^ = oo; i.Er/3, = 90»; 



and for elastic ones 



tang. j3i = — tang, a^, i.e. j3i = — ©i. 

After an inelastic body has impinged upon an inelastic obstacle, 
it moves on with the velocity Cj sin. a^ in the direction of the tan- 
gent plane. When an elastic body has impinged upon an elastic 
obstacle, it moves on with its velocity unchanged in the direction 

S G, which lies in the same plane as the normal iV^ iV and the 
original direction X S, and makes with the normal the same angle 

G S N that the direction of motion before the impact made with 

it. The angle X 8 N, formed by the direction of motion before 
the impact with the normal or perpendicular, is called the a7igle of 
incidence (Fr. angle d'incidence ; Ger. Einfalls-winkel), and the angle 
G S N, formed by the direction of motion after the impact with 
the same, is called the angle of reflexion (Fr. angle de reflexion ; Ger. 
Austritts- or Reflexions winkel) ; we can therefore assert that when 
the impact is perfectly elastic, the angles of incidence and of reflexion 
he in the same plane as the normal and are equal to each other. 

When the impact is imperfectly elastic, the ratio V/i of the 
tangents of these angles is equal to the ratio of the velocity pro- 
duced by the expansion to the velocity lost by the compression. 

By the aid of this law we can easily find the direction in which 



§ 341.] 



THE THEORY OF IMPACT. 



685 



Fig. 558. 



a body A, Fig. 558, must strike against an immovable obstacle 
£ B, when we wish it to take a given direction S Y after the im- 
pact. If the impact is elastic, we let fall 
from a point Y of the given direction 
a perpendicular Y upon the normal 

N N and prolong it until the pro- 
longation Yi is equal to the per- 
pendicular itself; S Yi is then the 
direction in question ; for, accord- 
ing to the construction, the angle 

WS Y,= NSY, If the impact is 

imperfectly elastic, we must make Y], — V^i » Y', then Yi 8 
is the required direction, for 

tang. ^1 _ Yi _ ^- 
toTig'.J: ~ 0~Y "~ ^' 
If we let fall the perpendicular Y R upon the line S R parallel 




= i/^ 



to the tangent plane and make the prolongation R X = y - R Y, 

/S'X will be, as we can easily see, the required direction of incidence. 
Remark. — The principal application of the theory of oblique impact 
is to the game of billiards. See '' Theorie Mathematique des effets du jeu de 
billard, par Coriolis." According to Coriolis, when a billiard ball strikes 
the cushion the ratio of the velocity of recoil to the velocity of impact is = 
0,5 to 0,6 or ^ is = 0,5' = 0,25 to 0,6' = 0,36. By the aid of these values 
the direction, in which a ball A must strike the cushion B B when it is 
to be thrown back towards a point Y, can be determined. We let fall 
from Fthe perpendicular TB to the line of gravity parallel to the cushion, 

prolong the same a distance B X = y - = i^ to ^^ of its length, and 

draw the line Y^ X; the point of intersection B is the point towards which 
the ball must be driven, when we wish it to rebound towards Y. The twist 
of the ball causes this relation to vary somewhat. 

§ 341. Friction of Impact. — When oblique impact occurs, 

the pressure between the colliding bodies gives rise to friction, in 
consequence of which the components in the direction of the tan- 
gent plane are caused to vary. The friction F of impact is deter- 
mined in the same way as that of pressure. If P denote the 
pressure of impact and the coefficient of friction, then F — (\i P, 
It differs from the friction of pressure in this only, that, like the 
impact itself, it acts but for an instant. The changes in velocity 



686 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 341. 



produced by it are not, however, immeasurably small ; for the 
pressure P during impact (and therefore the portion P of it) is 
generally very great. Denoting the impinging mass by M and the 
normal acceleration produced by the force of impact P by^, we 
have 

P = Mp and F = (j) M p, 
and also the retardation or negative acceleration of the friction 
during the impact 

I.E. times that of the normal force. Now the duration of the ac- 
tion is the same for both forces ; therefore the change of velocity pro- 
duced hy the friction is times the change of the normal velocity 
produced hy the impact. 

If a mass M falls vertically upon a horizontal sled, and if the 
velocity c of this mass is entirely lost by the collision, the retarda- 
tion of the motion of the sled, whose mass is J/i, is 

F _ Mp 
if + M, ~ WVM,' 
and consequently the loss of velocity is 

(j) M 



V = 



,T C. 



Fig. 559. 



M+ M, 

Morin has proved the correctness of this formula by experiment 
(see his Notions fondamentales de Mecanique). 

If a body strikes an immovable mass B B at an angle a, Fig. 
559, the change in the normal velocity is, according to the last 
paragraph, 

w = c COS. a {1 i- Vfi) ; 
hence the variation produced in the 
tangential velocity is 

= (j) w = (p c (1 4- ^fi) COS. a. 
After the impact the component csin.a 
becomes 

c sin. a — (j) c {1 + Vfi) cos. a 
= [sin. a — (f) COS. a (1 + Vij,)] c; 
for perfectly elastic bodies it is 

= {sin. a — 2 cos. a) e, 
and, on the contrary, for inelastic bodies it is 

=: {sin. a — COS. a) c. 




§ 341.] THE THEORY OF IMPACT. 687 

The friction very often causes the bodies to turn around tlieir 
centres of gravity, or if, before the impact, a motion of rotation ex- 
ists, it is changed. If the moment of inertia of a round body A in 
reference to its centre of gravity 8\^ — M ¥, and if its radius 8 G 
— a, we have the mass of the body reduced to the point of tan- 
gency G 

_M¥ 

and therefore the acceleration of the rotation produced by the fric- 
tion i^is 

_ F _ ^ ^^_ — rh — 

^' ~ M F:a' ~ WJ^T^' - "P P - ^^> 

and the corresponding change of velocity is 

tv^ = (^ — . w = (f) ^ {1 + Vii) c COS, a, 

a^ a^ 

For a cylinder -^ = 2, and for a sphere -^^ = #, therefore, it fol- 
lows that the changes of velocity of rotation of these round bodies, 
produced by impact against a plane, are 

Wi=^%(p {1 + V^) COS. a and w = | (1 --i- Vii) cos. a. 

Example. — If a billiard ball strikes the cushion with a velocity of 15 
feet, in such a manner that the angle of incidence a = 45°, what will be 

the conditions of motion after the impact ? Putting for V/^ its mean value 
0,55, we have the normal component of the velocity after the impact 

= — V/I. ccos. a = — 0,55 . 1^ .COS. 45° = — 8,25 . Vf = — 5,833 feet, 

and assuming, with Coriolis, <p = 0,20, we obtain the component of the ve* 
locity parallel to the cushion, which is 

= c dn. a — <i> (1 + sfjj) c COS. a = (1 - 0,20 . 1,55) . 10,607 = 0,69 . 10,607 
= 7,319 feet, 

and consequently for the angle of reflection we have 

7 319 
tang. (3 = ^^ = 1,2548 or /? = 51° 27'; 

hence the velocity after impact is 

5,833 
= ^^^1-27' = ^'^^^ ^''^' 

The ball also acquires the velocity of rotation 



688 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 343. 



10. 1,55 . 10,607 = 8,320 feet 
about its vertical line of gravity. 

Since the ball does not slide, but rolls upon the billiard table, we must 
assume that, besides its velocity c = 15 feet of translation, it has an equal 
velocity of rotation, and that this can also be resolved mto the components 

ccos. a = 10,607 and c sin. a = 10,607. 
The first component corresponds to a rotation about an axis parallel to the 
axis of the cushion, and becomes 

c COS. a — § (j) (1 + V/^) ccos.a = 10,607 — 8,220 = 2,387 feet ; 
the other component c sin. a = 10,607 feet corresponds to a rotation about 
an axis normal to the cushion and remains unchanged. 

§ 342. Impact of Revolving Bodies. — If two lodies A and 
B, Eig. 560, revolving around the fixed axes G and K, impinge upon 

one another, changes of velocity take 
place, which can be determined from 
the moments of inertia M^ h^ and 
J/2 ^2' of these bodies in reference to 
their fixed axes by the aid of the 
formulas found in the preceding para- 
graphs. If the perpendiculars G H 
and K L, let fall from the axes of ro- 
tation upon the line of impact, be 
denoted by a^ and a<i, we will have the 
masses reduced to the extremities H and L of these perpendicular 



Fig. 560. 




to the line of impact = 



and 



substituting these values 



for Ml and M^ in the formula for central impact, we obtain the vari- 
ations of velocity of the points H and L (§ 338). 



Ci — t'l = (ci — C2) 
= (ci — C2) 

V^ — c^ = (ci — Ca) 
= (ci — C2) 



.¥2 h^ ' ai 



My ki : ai + M^ Tc^ : a^ 
J/2 Tc^ a^ 



(1 + ^rt 



J/i Icy^ a^ + J/2 K' cii 
M, h,^ : a,' 



Ml ki : tti^ + J/2 K' ' a<i. 
Ml h' a,' 



(1 + V fi) and 

(1 + ^) 



(1 + ^), 



Ml k^ a,' + M, k,' a^' 
in which Ci and Cg denote the velocities of these points before the 
impact. 

To introduce the angular velocities, let us denote the angular 
yelocities before the impact by ei and £3 and those after the impact 



§342.] 



THE THEOEY OF IMPACT. 



689 



by Wi and Wg, thus we obtain Ci = a^ Cj, c^ — a^ eg? ^i = «i ^i and 
v^ — act Wsj and the loss of velocity of the impinging body is 



£1 — a>*i = ax («i £1 — «2 62) 



M, h,' 



M, h' a,' + M, h,^ a^ 
and the gain in velocity of the impinged body is 



0), 



— £3 = «2 {a\ £1 — ^2 £2) 



ifi ^1^ a,' + Jfs A^s' «i' 
The angular velocities after impact are 



(1 + ^ /^), 



(1 + V~^). 



Wi = fil — «^i («i fii — «2 £2) (1 + ^n^) 



M.K' 



-2 "^2 



and 



(i) 



= £2 + «2 («^1 £1 — «2 £2) (1 + i^ ft) 



M,Jc,' a,' ■\- M^k,' a,' 



2-^2 . ^2 V^I ^1 ^2 ^v v^ . ' r/ J^^ ^. ^. ^ J^^ ^. ^,' 

If both bodies are perfectly elastic, we have fi = 1, or 

1 + 1/77 = 2, 
and if they are inelastic, ft = 0, or 

1 4- V^ = 1. 
In the latter case the loss of vis viva occasioned by the impact is 

Mx hi' . M, h,^ 



= {CL\ £1 — «2 £2)'' • 



ifi h,^ a,' + M, h^ af 




Example. — The moment of inertia of the shaft A G, Fig. 561, in refer- 
ence to its axis of rota- 
Fi<^- 561. tion, G is 

= ifj h^^ = 40000 : g, 
and that of the tilt ham- 
mer B K in reference to 
its axis K is 

= 150000 : ff, 
the arm G C of the shaft 
is two feet and that K G 
of the hammer is 6 feet, and the angular velocity of the shaft at the mo- 
ment it impinges upon the hammer is = 1,05 feet ; how great is the velocity- 
after the impact and how much mechanical effect is lost by each blow, sup- 
posing both bodies to be completely inelastic ? The required angular 
velocity of the shaft is 

4 . 1,05 . 150000 . /. 60 \ , 

-J = 105 (1 - 204) = ^'^^ • ^'^^^ 



wi = 1,05 — 



40000 . 36 4- 150000 
= 0,741 feet, 

»nd that of the hammer is 

2.6. 1,05 . 4 , G C 

^ — 2oi — ^^^^ = "1 • zc = ^'"^^^ • ^ = ^'^^"^ ^®** 
44 



690 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 343. 



I.E., three times as small as that of the shaft. The loss of mechanical effect 
for each impact is 

(2.1,05^) 40000.150000 _ _ .^ 600000 

• 40000 . 36TT50000 . 4 "~ ^'^"^^^ ^^'^^' ' Tu'+^ 



A = 



^ff 



= 0,0155 . 4,41 



150000 10253,25 



51 



51 



= 201,05 foot-pounds. 



§ 343. Impact of an Oscillating Body.— If a body A, 
Fig. 562, which has a motion of translation and is 
unretained, impinges upon a body B C K^ movable 
around an axis K, we can find the velocities after 
impact by substituting in the formulas of the pre- 
ceding paragraph instead of a^ £i and a^ Wj the ve- 
locities of translation Ci and v^ and instead of 

^ M h"" 

the mass Jfj of the first body ; the other no- 




tti 



tations remain unchanged. The velocity of the 
first mass after the impact is therefore 



V, 



Ci — (Ci — «2 £2) (1 + 1/ |(X) . 



M, Jc,' 



and the angular velocity of the second is 

W2 -— Sc, + a.2 (ci — «o e.) {1 + V fj) . 



M, a,' + M, h? 

M, 



M, a,' + M, Tcf 
If the mass M^ is at rest, or if e.^ = 0, we have 



and 



V, 



0)o 



Ci — Ci{l + V fl) . 



C,{1 + V fl) . 



M, a,' + M, Tc^ 
M^a^ 



M, a,' + M, kf 

If i/i is at rest, i.e., if the oscillating mass impinges upon it, 
we have Ci = 0, and hence 

v^ — a^ 62 (1 + \^) . 
and 



M, a,' + M, k^ 



6)« 



Co ll 



(1 _ (1 + |/ ^) 



M^a^ 



.)■ 



M^ a^ + J/2 ^'2 

The velocity, which is imparted to a mass at rest by another 
by a blow, depends not only upon the velocity of the blow and the 
masses of the bodies, but also upon the distance K L — a^^X which 
the direction of the impact is situated from the axis K of the body 
which is capable of rotation. If the free body impinges upon the 
oscillating body, the angular velocity of the other becomes 



^343.] THE THEORY OF IMPACT. 691 

and if the oscillating body strikes against the free one, the latter 
acquires the velocity 

V, — t., (1 + r //) 



31, a^ + Jig ^• 



2 



both velocities increase, therefore, when 

a, 1 

or 



a 

increases, or 31, a^ + M^ — decreases. 

a.2 

Substituting for a^, a ^ x,x being very small, we obtain for 

the value of last expression 

31, (a±x) + — --^ ^3r,a±3f,x+ — ^-^ 1 qz - + — q= . . . , 
^ a ± X a \ a a I 

or, since the powers of x are very small, 

31, k,' ( ^^ 3f,k,'\ 

a \ a^ / 

Mo k ^ 
'N'ow if a is to correspond to the minimum value of 3f, a^ H ^^, 

/ 31, k.^\ 
the member ± [31, ^— ) x must disappear ; for its sign is 

different, Avhen a is increased a quantity (a;) from what it becomes, 
when a is decreased by a quantity {— x)\ hence we must have 

/ ^^ 31, ko\ 

J/' k ^ 

= 3I„ and consequently 




a 

a - 



^ ~3i,~ - ^' ^ m; 



Now if one body strike against the other at 
this distance (a), the latter assumes its maximum 
velocity, which is 



l)c.,^.{l + Vi.)-}-\/-^ = (l + Vii) 



2 k, ^ 3f, ' ' ^ 2 a 

when the oscillating body is impinged upon; and 

2)v, = i k, e, (1 + 4^) /J]= (1 +4/7) \~, 
when the free body receives the blow. 



• 



692 GENERAL PRINCIPLES OF MECHANICS. [§343. 

The extremity L of the distance or lever arm a = h.^ \' ~-^ 

which corresponds to the maximum velocity, or the point, where 
the latter line intersects the line of impact, is sometimes, though 
incorrectly, called the centre of percussion ; a more correct term 
would be the point of percussion. 

We should be careful not to confound it with the centre of per- 
cussion (§ 313), whose distance from the axis of rotation is ex- 
pressed by the equation _ ifs ^^2 _ ^2^ 

^ " "if, s ~~ T' 
in which s denotes the distance of the centre of gravity of the mass 

ifs from the axis of rotation. If the direction N N oi the impact 
of the masses Mx and M^ passes through the centre of percussion, 
the reaction upon the axis of rotation becomes = 0. 

In order, for example, to prevent a hammer from jarring, i.E. 
reacting upon the hand, which holds it, or upon the axis, about 
which it turns, it is necessary that the direction of the blow shall 
pass through the centre of percussion. 

If a suspended body K B is struck by a mass M^ with force F 

at the point of percussion, or at a distance a = Jc^y -~ from the 

axis Ky the reaction upon the axis is 

P, = P^Rz=^P-kM^s (see § 313). 

Since P = ^ — ^- , we have the angular acceleration k = -,, ^ ^ 
a Mi ICi 

H^ s a 
and K M^s — - .J. -.^ P ; hence the required reaction is 

Example — 1) The centre of percussion of a prismatical rod G A, Fig. 

564, Tvhich revolves about one of its ends, is 
Fig. 564. at a distance 

JC = a =\^ = ^r = % C A 
1 ^ ^ 



3 At ."1 "^^^^^^ from the axis. Now if we grasp the rod at 

^ one end and strike with the point 0, which is 

at the distance C = | G A^ upon an obstacle, we will feel no recoil. 

The point of percussion, on the contrary, is at a distance r y „ i^ from 
(7, and if the mass of the body struck M^ = ifg, we have this distance 

= — = 0,5774 r. The rod G A must therefore strike a mass at rest at this 

V3 



344.] 



THE THEORY OF IMPACT. 



693 



distance from (7, when we wish to communicate the greatest possible ve- 
locity to the latter. 

2) The distance of the centre of percussion of a parallel opipedon 

B D E, Fig. 565, from an axis X X, which is parallel to four of its sides 
and is at a distance S A = s from the centre of gravity, and about which 
the body rotates, is 



Fig. 565. 



a = 




d denoting the semi-diagonal G I> of the sides, 

through which the axis XX passes (§ 287). If the 
force of impact passed through the point of per- 
cussion, we would have 



a 



=^.l/'t = i/o 



^2 _ , //,. , , ,,.^ ^2 



1 1 fP\ ? 

and the reaction upon the axis would be 

§ 344. Ballistic Pendulum. — The principles discussed in the 
preceding paragraphs are applicable to the theory of the ballistic 
pendulum of Eobins (Fr. pendule ballistique ; Ger. ballistische Pen- 
del). It consists of a large mass M, Fig. 566, which is capable of 
YiQ 5gg turning around a horizon- 

tal axis C. It is set in os- 
cillation by means of a 
cannon-ball, which is shot 
against it, and serves to 
determine its velocity. In 
order to render the im- 
pact as inelastic as possi- 
ble, upon the side where 
the ball strikes, a large 
cavity is made, which from 
time to time is filled with 
fresh wood or clay, etc. 
The ball remains, there- 
fore, after every shot, 
sticking in this mass, and 
oscillates togethci- with 
the whole body. In order 
to determine the velocity 



-"N- 




694 GENERAL PRINCIPLES OF MECHANICS. [§ 344, 

of the ball, it is necessary to know the angle of displacement ; to 
determine this angle, a graduated arc B D is placed under it, along 
which a pointer, placed directly below the centre of gravity of the 
pendulum, moves. 

According to the foregoing paragraph, the angular velocity of 
the ballistic pendulum, after the impact of the ball, is 

_ M^ a^ Ci 

Ml denoting the mass of the ball, M^ h^ the moment of inertia 
of the pendulum, c^ the velocity of the ball and a^ the arm O G oi 

the impact or the distance of the line of impact N N from the axis 
of rotation. If the distance C M of the centre of oscillation M of 
the entire mass, including the ball, from the axis of rotation C, i.E. 
if the length of the simple pendulum, oscillating isochronously 
with the ballistic one, = r, and the angle of displacement E C D 
= a, the height ascended by a pendulum oscillating isochronously 
will be 

(2 
sin. ^J ; 

hence the velocity at the lowest point of its path is 

V — V'^g h — % V g r sin, -, 
and the corresponding angular velocity 

V ^ i/g . cb 
0) = - = 2 y - . sm. f: ; 
r T 2 

equating these values of the angular velocity, we have 

Now, according to the theory of the simple pendulum, 
_ moment of inertia _ M^i ai + M^ h^ 
statical moment (J/i + M^ s ' 

s denoting the distance G 8 oi the centre of gravity from the axis 
of rotation ; hence it follows that 

Jfi ai + M^ hi' = {M, -\- M,) s r and 

Ci = 2 I r— ) . — V q r . sm. -. 

If the pendulum makes 7i oscillations per minute, the duration 

of an oscillation is 

./T" 60" . ^, . ,/ 60" . g 

Try — r= — and thereiore y a r = -; 

g 71 ^ n n 

hence the required velocity of the ball is 



§ 345.] 



THE THEORY OF IMPACT. 



695 



Ci = 



M, + M, 120 gs 



a 
sm. ^. 



Fig. 567. 
B D 



Ml TITT a^ 

Example. — If a ballistic pendulum weighing 3000 pounds is set in os- 
cillation by a 6-pound ball shot at it, and the angle of displacement is IS"", 
if the distance 5 of the centre of gravity from the axis = 5 feet and the 
distance of the direction of the shot from this axis is = 5,5 feet, and, 
finally, if the number of oscillations per minute i& n = 40, the velocity of 
the ball, according to the above formula, is 

3006 120 . 32,2 . 5 . ^,, 501 . 3864 . si7i. T 80' ^,^^ ^ ^ 

^ = -^ • 4o73;i4r675-;5 ^^^- '^ = — 44:^,i4iT — = '^'^ ^'''' 

§ 345. Eccentric Impact. — Let us now examine a simple 
case of eccentric impact, where the t2vo masses are perfectly free. 
If two bodies A and B E, Fig. 567, strike each other in such a 

manner that the direction N N oi the impact 
passes through the centre of gravity ^S'l of one 
body, and beyond the centre of gravity S of 
the other, the impact will be central for one 
body and eccentric for the other. The action 
of this eccentric impact can be found accord- 
ing to the theorem of § 281, if we assume, in 
the first place, that the second body is free 
and that the direction of impact passes 
through its centre of gravity S, and, in the 
second place, that this body is held fast at 
the centre of gravity, and that the force of impact acts as a rotating 
force. Now if c^ is the initial velocity of ^, c that of the centre of 
gravity of B E, and if the two velocities become v^ and v, we have, 
as in § 332, M, v, -\- M v =: M^ c^ + Mc. If, further, £ is the 
initial angular velocity of the body B E, in turning about the axis 
passing through its centre of gravity perpendicular to the plane 

iV^ JV S, and if, in consequence of the impact, this becomes w, de- 
noting the moment of inertia of this body in reference to S by 
Mk"", and the eccentricity or the distance S K of the centre of 
gravity 8 from the line of impact by s, we have 

Mie ^^ M¥ 

. s (^ — M^ c^ + 




M, V, + 



s s. 



s - ' S' 

If the bodies are inelastic, both points of tangency have the 
same velocity after impact, then v^ = v -h s o). Determining from 
the foregoing equations v and o) in terms of v^, and substituting the 
values thus obtained in the last equation, we obtain 
M^ {c,-v,) , ^ , M, s'ic.-v,) 



Vi = 



M 



+ c + 



M¥ 



+ 5 e, 



696 GENERAL PRINCIPLES OP MECHANICS. [§346. 

from which we determine the loss of velocity of the first hody 

M¥ (g, — c - se) 
Ci - Vi = 



(M, + M) F + M, s" 
the gain in velocity of translation of the second 

J/i h' (cj — c — s e) 



V — c = 



and its gain in angular velocity 

Ml s (ci — c — s e) 
0) — e = ^ ' 



{M, + M)¥ -V M, s'' 
When the impact is a perfectly elastic one, these values are 

doubled, and when it is imperfectly elastic, they are (1 + V^) times 
as great. 

Example. — If an iron ball A, weighing 65 pounds, strikes with a ve- 
locity of 36 feet the parallelopipedon B E, Fig. 567, which is at rest and 
is made of spruce, if this body is 5 feet long, 3 feet wide and 2 feet thick, 

and if the direction of impact iViVis at a distance S K = s = If feet from 
the centre of gravity S, we obtain the following values for the velocities after 
the impact. If the specific gravity of spruce is = 0,45, the weight of the 
parallelopipedon is = 5 . 3 . 3 . 62,4 . 0,45 pounds = 842,4 pounds. The 
square of the semi-diagonal of side B D F parallel to the direction of the 
impact is 

(1)^ + (F = 7,25, 
whence (according to § 287), 

¥ = ^ . 7,25 = 2,416, 

g M¥ =^ 842,4 . 2,416 = 2035,2, 
and g {M^ + M)¥ = 907,4 . 2,416 = 2192,3; 

hence the velocity of the ball after the impact is 

M¥c, ^/, 2035,2 



v^ = e. 



_ / 2035,2 \ 

= 36 ^^ 1 - 2192^3 + 65 1,75V 



= 36 (l - 1^) = 36 . 0,149 = 5,364 feet, 

and that of the centre of gravity of the body struck is 

M^ ¥ Ci 157,08 . 36 _ 



V = 



2,364 feet ; 



(Jfi + M)¥ + ifi s^ "" 2391,4 
and finally the angular velocity is 

^.sc, _ 113,75_^ _ 

" = (if, + M)h^ + M, s^ - 2391,4 - ^' ''^ '^^'• 

§ 346. Uses of the Force of Impact.— The weight of a body 
is a force which depends upon its mass alone and increases uni- 
formly with it ; the force of impact, on the contrary, increases not 
only with the mass, but also with the velocity and with the hard- 
ness of the colliding bodies (see § 336 and § 338), and it can therefore 
be increased at will. Impact is consequently an excellent method of 



346.] 



THE THEORY OF IMPACT. 



697 



obtaining great forces with small masses or weights, and it is very 
often made use of for breaking or stamping rock, cutting or com- 
pressing metals, driving nails, piles, etc. On the other hand, im- 
pact occasions not only a loss of mechanical effect, but also causes 
the different portions of the machine to wear rapidly or even to 
break, and the durability of the structure or machine is seriously 
affected by it. For this reason it is necessary to make the dimen- 
sions of those parts of the machine larger than when the latter are 
subjected to extension, compression, weight, etc., without impact. 

If a rigid body A B, Fig. 568, strikes upon an unlimited mass 
C D C of soft matter, it compresses the latter with a certain force, 

whose mean value P is determined by means 
of the depth of the impression K L — s, 
when we put the work done P s during 
the compression equal to the energy of the 
mass of the striking body. If M be the 
mass, or G — g M the weight, of this body 
A B and v the velocity with which it strikes 
upon CD C, we will have 

IMv'^^ G, 

and the required force with which the soft matter will be com- 
pressed is 

^ s % g s 

Dividing this force by the cross-section of the body F, we obtain 
the force with which each unit of surface of the soft material is 
compressed and which such a unit can bear without giving way, 

_ P _ v^ G 

^ ~ F ~ ^'F~s 

For safety we only load such a mass with a small portion of p, 
for example with one-tenth part ( — ). 

The body M acquires its velocity v by being allowed to fall freely 




from a height h = ^^. If we substitute this height, instead of ^, 



^'j 



in the foregoing formula, we have 



P = - 



Gh 



or for the unit of surface p 



Gh 
Fs 



698 



GENERAL PRINCIPLES OF MEGHAN 



L^ 347. 



The force or resistance P, with which soft or loose granular 
masses oppose the penetration of a rigid body A B, is generally 
variable and increases with the depth s of the penetration. In 
many cases we can assume it to increase directly with s, I.E., that 
it is null at the beginning and double at the end what it is in the 
middle. Now since the value of P, deduced from the above 
formula, is the mean value, the resistance or proof load Pi of soft 
materials is twice as great as the value P obtained by the formula, 



Example. — If a commander A B^ Fig. 568, whose weight G^ = 120 lbs. 
falls upon a mass of earth from a height h =^ ^ feet, and if the latter is 
compressed |- an inch by the last blow, a surface of this material equal to 
the cross-section of the stamper will support a weight 



P = 



Gh 120 . 4 



1 

4 8 



23040 pounds. 



Fig. 569. 




Now if the cross-section F of the commander is f square feet, the force 
per square foot supported by this mass of earth would be 

P 23040 
p = -^ = —^ = 18432 pounds ; 

instead of which, for the sake of safety, we should take but ^^ JP — 1843,2 

pounds. 

§ 347. Pile-driving. — If we drive piles 
such as A B, Fig. 569, into earth or any 
other soft material C D C, we increase its 
resistance much more than we would by 
simply stamping it. Such piles (Fr. pieux ; 
Ger. Pfahle) are from 10 to 30 feet long, 8 
to 20 inches thick, and are provided with 
an iron shoe B. The body M, the so-called 
ram (Fr.mouton; Ger. Eammklotz, Eamm- 
bar or Hoyer), which is allowed to fall from 
3 to 30 feet upon the top of the pile, is gen- 
erally made of cast iron, more rarely of oak, 
and weighs from 5 to 20 hundred weights. 
If the ram falls the vertical distance h, the 
velocity with which it strikes the pile is 




and if its weight = G and that of the pile 



§347.] THE THEORY OF IMPACT. 699 

= 6^1, we have, when we suppose that both bodies are inelastic, the 
velocity of the same at the end of the impact (see § 332) 

- ^^ 
hence the corresponding height due to the velocity is 

'^g~\G^Gj'^g~\G^- gJ 

Now if the pile sink during the last blow a distance s, the re- 
sistance of the earth and the load which the pile can support is 

2gs ^ s G + Gi 

or more correctly, since the weight G -\- Gi ot the pile and ram act 
in opposition to the resistance of the earth, ♦ 

In most cases G -\- G^ is so small, compared to F, that we can 
neglect the latter part of the formula. 

If the weight G^ of the pile is much smaller than the weight G 

of the ram, we can write 

Gc 

and simply P = - G. 

The foregoing theory suffices in practice, when the resistance P 
is moderate and, consequently, the depth s of the impression is 
not very small ; for in that case the compression of the pile, etc., 
can be neglected. If, on the contrary, the resistance P is very 
great and, consequently, the depth s of the impression very small, 
the compression a of the pile can no longer be regarded as null, and 
must therefore be introduced into the calculation. 

The pile of course does not begin to sink until the force of 
impact has become equal to the resistance P of the earth. Now 

if II = — - — and 11^ = -— — ^ denote the hardness of the ram and 

that of the pile (in the sense of § 336), the sum of the compressions 
of the two bodies, when the force of impact is P, is 

and the mechanical effect expended in producing this com- 
pression is 

T _P<y _ C^ i\p» 



700 GENERAL PRINCIPLES OF MECHANICS. [§347. 

JS"ow if this first impact of the two bodies causes the velocity c of 

the ram to become v. its mass if = — performs the work 

9 

hence we can put 



from which we obtain 



(c^ - v\ ^ /I 1 \ P' 



v' c' /I 1 \ P^ 



2a \H 



, = /,._,,(!,. ^) 



2g 2g MT If,/ 2 G' 
consequently the velocity of the ram, when the pile begins to pen- 
etrate the earth, is 

TG' 

We infer from this that a pile (and also a bolt or nail in a wall) 
will begin to enter the resisting obstacle when 

c^ /JL^ 2_\ p; 

2 g^^\JI '^ nj 2' 

or when the weight of the ram and its velocity have the proper re- 
lation to the resistance of the earth. During the penetration of 
the pile the force of impact and, consequently, the compression of 
the pile, etc., diminish as long as the velocity of the ram exceeds 
that of the pile ; when both attain a common velocity v, and the 
force of impact becomes a maximum, the bodies begin to expand 
again. During this expansion not only the velocity of the ram, 
but also that of the pile becomes gradually = ; the pressure be- 
tween the two bodies becomes again P, and consequently at the 



moment, when the pile ceases to penetrate, the whole energy 

^ 9 
G of the ram is consumed by the work 



\H ^ Hj 2 



expended in compressing, and by the work 

P s 
done in driving the pile to the depth 5. 

Hence we have 



^' G 



'^g 



-^^ = (i^i-)? + ^^' 



§347.] THE THEORY OF IMPACT. 701 

and therefore the load which corresponds to the depth of penetra- 
tion s is 

/I 1 \ P' 

If the compression (-^ + -^1 -^ is considerably smaller than 

the space s described by the pile, we can write simply 

' c' G Gh ,, 

F = = , or, more accurately, 

2g s s ' •'' 

Gh 



/I 1 \Gh' 

Comparing the work done in driving in the pile 
p Gh 

with the work done G h in raising the ram, we see that the former 

approaches the latter more and more as ( -^ + -^1 -^ becomes 

WW TP W 

smaller or as the hardness S= — = — of the ram aod that H^ = --^j—^ 

of the pile become greater, i.e. the greater the cross-sections F and 
Fi and the moduli of elasticity F and Fi of these bodies are and the 
smaller the lengths are. 

The action of the weights of the two bodies can be entirely neg- 
lected, since they generally form but a small portion of the resist- 
ance F. We can also neglect the energy, which the bodies possess 
in consequence of their elasticity (although the latter is imperfect) 
after the pile has come to rest ; for the body, which is thrown back 
by their expansion, is generally, upon falling again, incapable of 
overcoming F and setting the pile in motion. For safety's sake, 
the pile, which has been driven in, is loaded with only j^q part of 
the resistance P, just found, or perhaps with even less. According 
to some late experiments made by Major John Sanders, U. S. A., 
at Fort Delaware (communicated by letter) we can put, approxi- 
matively, the resistance 

3s' 
Example. — A pile, whose cross-section is 1 foot = 144 square inches, 
whose length is 25 feet = 25 . 12 = 300 inches, and whose weight is 1200 
pounds, is driven by the last tally of ten blows of a ram, weighing 2000 



m 



702 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 348. 



pounds and falling 6 feet = 73 inches, 2 inches deeper, what is the resist- 
ance of the earth ? If we neglect the inconsiderable compression of the 
cast iron and put (according to § 213) the modulus of elasticity of wood 
E^ = 1,560000, we obtain 

Zi 300 1 



(^^^) 



+ 



2F^E^ 3 . 144 . 1,560000 ~" 1497600" 
Now since G h — 3000 . 73 = 144000 inch-j)ounds and the depth of the pen- 
etration after one blow is s = ^ = 0,3 inches, we obtain for the determina- 
tion of P the following equation : 

^' + 0,3 P = 144000 or P^ + 399520 P = 315654400000. 

Resolving this equation, we obtain 



P = _ 149760 + V 338083457600 = 338177 pounds. 
According to Sanders' formula 

^^ ^^*««»= 340000, 



3 s ~ 0,6 
while the old formula, on the contrary, gives 

G^'Ji G GJi 3000 



^ ~ {G + G^)s~~ G-^ G^' s 
= 450000 pounds. 
From P = 338177 pounds we obtain 



144000 



3300 • 0,3 



= #.730000 



[W -^ hJ 2 - 



76365 inch-pounds. 



and therefore the height from which a ram weighing 3000 pounds must 
fall in order to move the pile is 



h 



/I 1 \ P' 76365 

{w + H;)¥G = -2000 =38,3 inches. 



Fig. 570 
A 



348. Absolute Strength of Impact. — By the aid of the 

moduli of resilience and fragility (see § 206) we can 
easily calculate the conditions under which a prismatical 
body A B, Fig. 570, will be stretched to the limit of elas- 
ticity or broken by a blow in the direction of its axis. If 
G be the weight and c the velocity of the impinging 
body, the work done, when the prismatical body, whose 
weight we will denote by Gx, is struck, is 

^2 



L = 



2g ' G -h G,' 



Gi 



B 



or denoting the height due to the yelocity ^— by h, we 

have more simply 

G'h 



L = 



G-\- g; 



g348.] THE THEORY OF IMPACT. 703 

This mechanical effect is chiefly expended in stretching the rod 
A B, upon which the second body hangs ; if, therefore, H is the 
hardness, I the length, F the cross-section, E the modulus of elasti- 
city, P the force of impact and A the extension of the rod produced 

by it, we have 

PX P"" F E 

and consequently 

FE _ G'h 

from which the extension X of the rod, caused by this impact, can 
be easily calculated. 

If the rod is to be extended only to the limit of elasticity, we 
have, when A denotes the modulus of resilience (§ 206), 

L=^ A V= AFl, 

and therefore a th G^ 1^ 

A Fl 



G ■\- G' 



the velocity of impact c — V 2 g h, which is necessary to stretch it 
to the limit of elasticity, is determined by the height 

y G -\- Gi . 

h — — 775 — - . A Fl 

G' 

If we are required to find the conditions of rupture of the rod, 
we must substitute, instead of the modulus of resilience A, the 
modulus of fragility B. 

We see from this that the greater the mass of the rod is, the 
greater is the blow it can bear. Hence we have the following im- 
portant rule, that the mass of bodies subjected to impacts should 
be made as great as possible. 

Since G and G^ fall the distance X during the impact, it is more 
correct to put 

•^ ~ Q -I- Q + (^ + ^i) ^f 

or for the case, when the limit of elasticity is reached, 

in which r = ^ expresses the extension corresponding to the limit 

of elasticity. 

If, finally, we wish to take into consideration the mass and 
weight Gi of the rod, we have, since its centre of gravity sinks but 

•2 ^f 



704 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 348. 



A F 



G' 



.^ + (6? + G,-^l G,)a. 



6r + Gx + G^ I 

We have a similar instance of the action of impact, when a 

G Q 

moving mass Jf = — , Fig. 571, puts another mass ifi = — in mo- 

9 g 

Fig. 571 



^-c 




tion by means of a chain or rope. If c is the velocity of M at the 
moment, when the chain is stretched, v the velocity with which 
both bodies move after the impact, we have again 

Mc G G, 



V 



M -r M,~ G + G,' 
while, on the contrary, the work expended in stretching the chain is 

~M + Mj 
. Ti. 



MM, c" G G, 






M + M,' 2 G + G, 

If, therefore, this chain, etc., is to be stretched only to the limit 
of elasticity, we must put 

G Gi 



AFl = 



-rT- h 



G + G, 
F denoting the cross-section and I the length of the chain. 

Example — 1) If two opposite suspension-rods of a chain bridge sup- 
port a constant weight of 5000 pounds, which is increased 6000 pounds by 
a passing wagon, if the modulus of resilience A of wrought iron is 7 inch- 
pounds and if the length of the suspension-rods is 200 inches and their 
cross-section 1,5 square inches, we have the dangerous height of fall 



Ji = 



AFljO + G^) _ 7 . 3 . 1,5 . 200 . 11000 _ 7.11 _ 77 



G' 



36000000 



60 



= 777T = l^^S inches. 
dO ' 



If the wagon passes over an obstacle 1,3 inches high, the suspension-rods 
would already be in danger of being stretched beyond the limit of elas- 
ticity. 

2) If a full bucket or loaded cage in a shaft is not gradually set in mo- 
tion, but if by means of the rope, which has been hanging loosely, it is sud- 
denly brought to a certain velocity by the revolving drum, the rope wil] 
often be stretched beyond the limit of elasticity, and sometimes even 



§ 349.] 



THE THEORY OF IMPACT. 



705 



broken. If the mass of the drum and shaft, reduced to the circumference 

of the former, is if = — = , the weight of the full bucket or cage 

is G^ = 2000 pounds, and the weight of the rope = 400 pounds, then if 
the weight of a cubic inch of rope is = 0,3 jpounds, its volume will be 



-nil ^1 4^^ 

^^^x- o;3~ 



4000 , . . , 
— ^- cubic mches, 



and, finally, if the modulus of fragility of this rope is = 350 pounds, we 
have the height due to the velocity, which will break the rope, 

Qj^a^ 4000 100000 + 2000 1400000 102 



a a^ 3 • 100000 . 2000 3 * 200000 

= 238 inches = 19,83 feet, 
and, therefore, the velocity of the rope at the beginning of the strain is 



c = V2^A = V 64,4 . 19,83 = 35,74 feet. 

§ 349. Relative Strength of Impact. — The foregoing the- 
ory is also applicable to the case of a prismatic body B B, Fig. 
572, supported at both ends and exposed to the blows of a body 
A, which falls from the height A C = 7i upon its middle C. Let 

— = if be the mass of the falling body and M^ that of the body 

B B, reduced to its middle C, then the energy of the bodies after 
the impact is 



L = 



M' 



M 



2 ' M-\-M, 2g ' M + Mi 
Fig. 573. 



r'^9 = 



M 



M+M, 



Gh. 





The mass M^ of the beam B B can be determined in the follow- 
ing manner. Let G^^ be the weight, I half the length B D, Fig. 
673, of this beam, x the abscissa B N and y the corresponding 
ordinate N oi the curve, formed by ^ ^ at the moment of 
greatest flexure, and, finally, let a denote the maximum deflection 
CD of this curve. If we imagine B (7 toi be divided into n (an infi- 
nite number of) parts, the weight of an element of the rod will 
45 



706 GENERAL PRINCIPLES OF MECHANICS. [§349. 

C 
be — , and therefore the mass of an element of the rod, reduced 

n 

from iV^ to D, is 

n g \C D I n g (f 
But, according to § 217, 

whence it follows that the element of the mass of the rod is 



9G,x'(r -irx' + ~) 



4: n g r 

111 1 

Now if instead of x we substitute successively -, 2 -, 3 - . . . — 

"^ n n n n 

and add, etc., the values thus obtained, we obtain the mass of the 
rod B B, reduced to its middle (7, 

_ 9 6^1 / f' _ 2 72 ?! 1 1 n — -11 ^ 

If we substitute this value, we can put the work done by the 
impact 



M + M, ^ + ii Gi 

and obtain the condition of bending to the limit of elasticity (see 

§ 235), 

, Wl GMi 

A . 



3 e^ G + \\ g; 

If the beam is a parallelopipedon, we have 

G'h 



^ A V — 

^^ ^1 - a J, 17 a^' 



35 

and therefore 



. = ^i^tii^,orputti„gF. = f, 



n 



_AG,{G + 11 G) 



^y G' 

If we substitute B instead of A, the expression becomes 
. _ BG,{G + ilG,) 

^- ^jG' 

and gives the height, from which the weight G must fall in ordei 
to break the parallelopipedical rod. 



§ 350.] THE THEORY OF IMPACT. 707 

Example. — From what a height must an iron weight G = 200 pounds 

fall, in order to break by striking it in the middle a cast iron plate 36 

inches long, 13 inches wide and 3 inches thick, which is supported at both 

ends ? 

The modulus of fragility 

B = 14,8 inch-pounds 

(see § 211), and the volume of the plate is 

Fj = 5 A Z = 12 . 3 . 36 = 1296 cubic inches, 

and, since a cubic inch of cast iron weighs y — 0,259 pounds, its weight is 

G^ = 1296 . 0,259 = 335,7 pounds ; 

the required height is 

14,8 . 335,7 (200 + U . 356,4) ,^, . , 
7; — ! I. ^^ I °1 ^-Z — 18^ inch pq 

'' - 9 . 0,275 . 40000 " ^^* '''^^^'• 

§ 350. Mechanical Effect of the Strength of Torsion. — 

We can also investigate the action of impact in twisti^ig shafts. 
According to § 262 the mechanical effect which is required to pro- 
duce a torsion a in a shaft, whose length is I and the measure of 
whose moment of flexure is W, is 

_ P_oji _ a' , W C _ P' d" I 

2 ~ 2/ ~ 2 W C 
we can also put 

Tc ~e" ^^^^ ^ ^' 

€ denoting the distance of the most remote fibre from the neutral 
axis and 8 the strain in that fibre. 

If we substitute for 8 the modulus of proof strength T, and for 

rp rs n^ 

—^- the modulus of resilience A, we obtain the work to be 



2 (7 2 

performed m stretchmg the remotest fibre to the limit of elasticitv 

Wl 



L = A, ^^, 



and the mechanical effect necessary to rupture the shaft by wrench- 
ing, when we substitute for the modulus of resilience A the modu- 
lus of fragility B ; its value is 



e' 



7T T* 

For a cylindrical shaft W — -^r- and e — r, hence 

<> /i z z 

when V = TT r"" I denotes the volume of this shaft. 

For a shaft with a square cross-section^ the length of whose side 
is h, we have 



708 GENERAL PRINCIPLES OF MECHANICS. [§350, 



PF = — and « = 5 V i, 



6 
and consequently 

L=^hH = ^ FandXi = |- F. 

6 6 6 

If a revolving wheel and axle, whose mass reduced to the point 

of impact is if = — , impinges upon a mass ifi = — , which is at 

rest, with the velocity c, both will move on after the impact with 
the velocity 

^^ ^^ ( Q OOA\ ;i 

^ = m^-. — iTF = ^ 7T (see 8 334), and 

consequently the mechanical effect 

G -{- G,' 2g' 

which is expended in twisting the axle and bending the arms of the 
wheel, is lost (see § 335). 

But L is also the sum of the mechanical effects expended in 
producing the torsion of the axle and in bending the arms of the 
wheel, etc., I.E., 

e ei' 

when Ai denotes the modulus of resilience, Wi the measure of mo- 
ment of flexure and ei the distance of the exterior fibre from the 
neutral axis (see § 235) ; we can therefore put 

A Wl A, PFi ?i _ G G, ^ 

e' "^ 3 ei^ ~ G + G, 2 

W I V 
If the shaft is cylindrical, we have -^- = — , and if it is four- 

Wl V 
sided, we have — ^ = — , when V denotes its volume ; and for the 

6 O 

four-sided arm we have 7^-^ = ~, where F denotes the volume 

O 61 V 

of the arm. 

Hence for a cylindrical shaft we have 
A_ A, G G, ^ 

2 ^ 9 "^^ G + G, 2g' 
and, on the contrary, for ^four-sided shaft 

A^ A, G G, ^ 

3 "^ 9 ' ~ G + G, 2 g 



§350.] THE THEORY OF IMPACT. 709 

The volumes V and V^ have a certain relation to each other, 
which can be expressed as follows. The moment of flexure of the 
arms is equal to the moment of torsion of the shaft. 
Hence 

WT W,T, 



e 



, or 



16 ~ 3 |/2 ~ 6 ' 

jPand Ti denoting the moduli of proof strength for torsion and 
bending and d the diameter of a round, and b the length of the 
sides of a four-sided shaft, while hi is the thickness and bi the sum 
of the widths of all the arms of the wheel. 

But we have also V = —r— I = If I and Vi = bi hi k, and 



therefore 



'TTd'lA bi hi liAi_ G Gi & 



^^ ^ A«7^ _L ^I'^i^i-^i _ G Gi j^ 
-,t> lA + ^ - G + Gi'2g 

Now if the ratio v =: ^ of the dimensions is given, we can cal- 

fii 

culate the thickness d or h of the shaft of the thickness hi and the 
width bi of the arms by means of equations (1) and (2). We must 
introduce into this calculation 

1) for cast iron 

T^ 1906^ 

A, = 3,16 and ^ = ^= ^-y^glOOOO = ^'^^'^ ''"'^-^^'•' 

2) for wrought iron 

T' 59742 

A, = 6,23 and ^ = ^ = ^ _ qqoooqo = 1,983 inch-lba, 

3) and for wood, the mean value 

T^ 3952 

A, = 3,17 and ^ = ^ = ^-^^-^^^ = 0,133 ineh-lbs. 

Example. — Let the mass of the wheel, etc., of a tilt-hammer, reduced to 

the point of application of the cam, be M= pounds, and the mass of 

25000 
the hammer reduced to the same point be M = pounds, and let the 



710 GENERAL PRINCIPLES OF MECHANICS [§350. 

distance from tlie wheel to the ring, in which the cams are set, be Z = 15 
feet = 225 inches, and the length, of the arms of the wheel be Ix = 10 feet = 
120 inches. Now if the hammer, every time it is lifted, is struck with a 
velocity of 2 feet, how thick must the shaft and the arms of the wheel be 
made in order to sustain this impact without being damaged f If the shaft 
and arms are of wood, we have 



395 -— = 1000 



16 6 ' 

and if the number of arms is n = 16, we can put 

i^ = V .nh^ = 0,707 . 16 A^ = 11,3 . h^^ 
whence we obtain 



But 



, , ;/ 16000. 11,3 ^. , 
^=^^V 6.395 . --^>9'V 



^Al = 0,132 . 225 ^ = 11,66, 

o o 

^ ^1 ?! = i . 2,17 . 120 = 28,9, 



and also 



^^1 «' -.o nm^r. . 200000.25000 ^_^ 5000000 

12 . 0,0155 . 4 . or^nnnn . .....r. = 0,^44 



G+G^' 2g ' ' • '200000 + 25000 ' '225 

= 16533 inch-pounds ; 

hence we have the equation of condition 

(2,9)' . 11,66 h^^ + 11,3 . 28,9 h^^ = 16533, i.e., 

98,1 Ai' + 326,6 h^^ = 16533, 

hence the required thickness of the arm 

V16533 ^„, . , 
^1 =V 424^ = ^M inches 

the width of the arm 

5^ = 0,707 A 1 = 4,41 inches, 
and the thickness of the shaft 

d = 2,9 h^ = 18,1 inches. 

For the sake of security we make the dimensions considerably larger. 

Remark. — It is only of late years that much attention has been paid to 
the strength of impact. We find something in regard to it in Tredgold's 
work on the strength of cast iron, in Poncelet's " Introduction a la 
Mecanique Industrielle," and in Riihlmann's " Grundziige der Mechanik 
und Geostatik." The discussion in the latter work is based principally 
upon Hodgkinson's experiments on the resistance of prismatic bodies to 
impact, upon which subject an article by Bornemann is to be found in the 
■' Zeitschrift fiirdasgesammtelngenieurwesen" (the Ingenieur). 

The experiments of Hodgkinson agree essentially with the foregoing 
theory of the strength of impact ; they apply particularly to relative 



§350] THE THEORY OF IMPACT. 711 

strength, and were made in the following manner : large weights swinging 

like pendulums were caused to strike against rods supported at both 

(9:2 ^ 

ends. The formula L = 7; , which we found by assuming that the 

"" + t ""1 
impact was perfectly inelastic, was verified completely ; the mechanical 

effect L was found not to depend upon the nature of the colliding bodies. 
Equally heavy bodies of different materials (cast iron, cast steel, bell metal, 
lead) produced, when they fell from the same height, equal deflections of 
the same rod (of cast iron or cast steel) ; the deflections were almost ex- 
actly the same as those given by the theory for a perfectly elastic rod. 

FrNAL Remakk. — For the study of the Mechanics of rigid bodies, be- 
ffldes the older works of Euler, Poisson, Poinsot, Poncelet, Navier and 
Coriolis, and those of Whewell, Mosely, Eytelwein and Gerstner, the follow- 
ing are recommended : 

Duhamel, Cours de Mecanique, Paris, chez Mallet-Bachelier, 1854 ; 
Sohnke, Analytische Theorie der Statik und Dynamik, Halle, 1854 ; Broch's 
Lehrbuch der Mechanik, Berlin, 1854 ; Morin, Lecons de Mecanique pra- 
tique, Delaunay, Traite de Mecanique rationelle, Paris, 1856 ; Rankine, A 
Manual of Applied Mechanics, second edition, London, 1861 — a valuable 
work, too little prized in England. A translation of a new Monograph 
upon impact, by Poinsot, has lately appeared in the third year of Schlo- 
mich's Zeitschrift f iir Mathematik und Physik. 



SIXTH SECTION. 

STATICS OF FLUIDS. 



CHAPTER I. 

OF THE EQUILIBRIUM AND PRESSURE OF WATER IN VESSELS 

§ 351. Fluids. — We consider /m^Zs to be bodies composed of 
material points, whose coherence is so slight that the smallest force 
suffices to separate them from each other (§ 62). Many bodies 
which are met with in nature, such as air, water, etc., possess 
this distinguishing property of fluids in an eminent degree, while 
others, on the contrary, such as oil, tallow, softened clay, etc., pos- 
sess a less degree of fluidity. The former are called perfectly, and 
the latter imperfectly fluid, or viscous bodies. Certain bodies, as, 
E.G., dough, lie midway between the solids and the fluids. 

Perfectly fluid bodies, of which only we will treat in the discus- 
sion which is to follow, are at the same time perfectly elastic, i.e. 
they can be compressed by extraneous forces, and when these forces 
are removed, they reassume the primitive volume. But the amount 
of change of volume corresponding to a certain pressure is very dif- 
ferent for diflerent fluids ; while in liquids this change is quite un- 
important, in gaseous or aeriform fluids it is very great, and they 
are therefore called elastic or compressible fluids. On account of 
the slight degree of compressibility of liquids, they are treated in 
most of the researches in hydrostatics (§ 66) as incompressible or 
inelastic fluids. As water is the most generally diffused of all 
liquids and is the most generally employed in practical life, we 
regard it as the representative of all these fluids, and in the re- 
searches in the mechanics of liquids we speak only of water, with 



§352.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 713 

the tacit understanding that the mechanical relations of other 
liquids are the same. 

For the same reason, in the mechanics of elastic fluids we speak 
only of common atmospheric air. 

Remabk. — A column of water, whose cross-section is one square inch, 
is compressed by a weight of 14,7 pounds, corresponding to the weight of 
the atmosphere, about 0,00005 or one fifty millionth of its volume, while a 
column of air under the same pressure occupies but one-half of its primi- 
tive volume. See Aime " Ueber die Zusammendriickung der Fliissigkeiten" 
in Poggendorff's Annalen, Erganzungsband (to Vol, 72), 1848. According 

to the formula P = j F E {^ 204), we have, when P ~ 14,7 pounds, F — 

1 square inch and 7- = ttttt^t. = ^^^»^ , the modulus of elasticity of water 
^ I 100000 20000' ^ 

PI 
Ez=. —- = 14,7 . 20000 = 294000 pounds. 

J? A 

% 352, Principle of Equal Pressure. — The characteristic 
property of fluids, by which they are principally distinguished from 
solid bodies and which forms the basis of the theory of the equili- 
brium of fluids, is the capacity of transmitting the pressure exerted 
upon a portion of their surface unchanged in all directions. In solid 
bodies the pressure is transmitted only in its own direction (§ 86) ; 
if, on the contrary, water is subjected to pressure on one cide, the 
same pressure is exerted throughout all the mass of fluid and can 
consequently be observed at all parts of the surface. In order to 
conyince ourselves of the correctness of this law, we can employ 

an apparatus filled with water, like 
the one whose horizontal cross-sec- 
tion is represented in Fig. 5 74. The 
tubes A E, B F, etc., which are of 
the same size and at the same dis- 
tance above the base, are closed by 
pistons, which are easily movable 
and which fit the tubes perfectly ; 
the water will then press upon each 
of them, by virtue of its weight, ex- 
actly as much as upon the others. 
Let us for the present disregard 
this pressure and regard the water as imponderable. If we exert 
against one of the pistons A a certain pressure P, the water will 




714 



GENERAL PRINCIPLES OF MECHANICS. 



L^ 352. 



transmit the same pressure to the other pistons B, C, D, and to 
preserve the equilibrium or to prevent these pistons from moving 
backwards, an equal opposite pressure P (Fig. 575) must be exerted 
against each of the other pistons. We are therefore authorized to 
assume that the pressure P exerted upon a portion A of the surface 
produces a strain which is propagated not only in the straight 
line A C, but also in every other direction B F, D H, etc., upon any 
equally large portions C, B, D of the surface. 



Fig. 575. 



Fig. 576. 





If the axes of the pipes B F, C G, etc., Fig. 576, are parallel to 
each other, the forces acting on the pistons can be combined so as 
to give a single resultant ; if n is the number of the equally large 
pistons, the total pressure upon them will be 

P,^nP\ 
in the case represented in the figure 

P,^ 3 P. 
Now the aggregate area F^ of the surfaces B, C, I>, upon which 
the pressures are exerted, is also = n times the area F of one 

P P 

of the pistons ; w is therefore not only = -^, but also -^, or in 

general 

P ~ ~F ^ ^ ~ ~F 

Now if we cause the tubes B, C, D to approach each other, 
until they form, as m Fig. 577, a single one, and if we close the 
latter by a single piston, F^ becomes a single surface and P, is the 
pressure exerted upon it ; hence we have the general law : the 
2)ressures exerted hy a fluid upon the different parts of the walls of 
the vessel are proportional to the areas of those parts. 



§353.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 715 



Fig. 577. 



Fig. 578. 

K by 




This law corresponds also to the principle of virtual veloci- 
ties. If the piston A D — F, Fig. 578, moves a distance A A^ — s 

inwards, it presses 
the prism of water 
F s out of its tube, 
and the piston B E 
= F^ moves out- 
wards the distance 
B Bi = Si and leaves 
behind it the pris- 
matical space F^ Si. 
Now as we have 
assumed that water can be neither expanded nor compressed, its 
volume must remain the same after the pistons have been moved, 
or the increase F s must be equal to the decrease Fi s^. But the 
equation Fi Si = F s gives 

F ~ 6'i' 

and by combining this proportion with the proportion 
we obtain ^i __ "^ . 

hence the mechanical effect P^ s^ = P s (see § 83). 

Example.— It the diameter of the piston A D\sl\ inches and that of 

the piston B E is 10 inches, and if the pressure exerted by the former upon 

the water is 36 pounds, that exerted upon the latter piston is 

F, 10' 400 

P^=~^P= Y^, . 36 = — - . 36 = 1600 pounds. 

If the first piston moves 6 inches, the second moves but 

= ^L. = 0,135 inches. 



Pi 
P 



?1 
F 



«i -^^ 



s = 



400 



Remakk. — In the following pages we will meet mth many applications 
of this law, E.G., to the hydraulic press, water pressure engines, pumps, etc. 

§ 353. Pressure in the Water. — The pressure exerted by 

the particles of water against each other 
must be estimated in exactly the same 
manner as the pressure of the water against 
the wall of the vessel. The pressures upon 
both sides of any surface E C G^ which di- 
vides the water in a vessel B G H, Fig. 579, 
into two parts, when equilibrium exists, 
are equal. Now as a rigid body counter- 




716 



GENERAL PRINCIPLES OP MECHANICS. 



[% 353. 



Fig. 580. 




acts all forces whose directions are at right angles to its surface, 
the conditions of equilibrium will not be disturbed, when one-half 
E H oi the liquid becomes rigid, or if its limiting surface 

becomes a wall of the vessel. If the fluid 
half ^^ G in one portion G D ^ F, of 
the imaginary surface of separation EGG 
exerts a pressure P^ upon the rigid half 
E G H, the latter counteracts this pres- 
sure completely and will react with an equal 
^ti opposite pressure {— P^) upon G D = F^. 
Since the conditions of equilibrium will 
not be changed, when this mass of water 
E G H becomes fluid again, the latter will react with an equal 
pressure ( — P) upon the mass of water E B G\ hence the pres- 
sure of the water upon both sides of a surface G D = Fi^ also de- 
termined by the proportion 

P,_E\ 

P F' 

when all the water is pressed in a surface ^ ^ = i^ by a force P. 
Hence the pressure upon any giyen surface F^ in any arbitrary 
position is 

P - ^P 

The law of the transmission of pressure in water, expressed by 
the last proportion, is only applicable when we consider water as 
an imponderable fluid, and it must therefore be modified, when it 
is required to determine in addition the pressure arising from the 
weight of the water. If we imagine a part of the water in a vessel 
G D E, Fig. 581, to become rigid and to have the form of an infi- 
nitely thin horizontal prism A B, 
it is easy to see, that the pressures 
of the water, that remains fluid, 
upon the sides of the rigid part 
balances the weight G of the prism 
and that the horizontal pressures 
upon the vertical bases A and B 
of this part counteract each other. 
These pressures (P and — P) must 
therefore be equal and opposite to 
each other. Since the state of equilibrium is not changed, when 
A B again becomes fluid, it follows that the pressures of the 



Fig. 581. 




§353.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 717 



Fig. 582. 



water against the vertical eleraents A and B of the surface, whicli 
are situated m one and the same horizontal plane, must be equal 
to each other, and since the pressure upon an element does not 
change, when its inclination or direction changes, it follows that 
the water in a horizontal layer, as, e.g., G H, K L, etc., exerts the 
same pressure m all directions and in all positions. 

If we imagine a vertical prism A B, whose cross-section is infi- 
nitely small, to become rigid in the mass of water C H K, Fig. 582, 

we can conclude from the conditions of its equi- 
librium with the remaining liquid that the 
pressures exerted by the latter upon the vertical 
sides of the prism balance each other and that 
the weight G of the latter body is in equilibrium 
with the excess Pi — P of the pressure Px upon 
lower base B above the pressure P upon the 
upper base A. Hence P^ — P — G, i.e. the 
pressure P^ of the water upon any elementary 
surface B is equal to its pressure P upon an ele- 
ment A, of equal size and situated above it, plus 
the weight (x of a column of water A B, whose 
base is one or other elementary surface and 
whose height is the vertical distance between the 
two elements. According to what precedes this 
rule is not only applicable to two elements, 
situated vertically above one another, but can also be employed 
for determining the pressure upon the walls of the vessel ; for the 
two pressures P and P^ are transmitted unchanged in the hori- 
zontal planes G H and K L. Hence the pressure i? upon an ele- 
mentary surface B, K ov L of the horizontal plane K L is equal to 
the pressure P upon an equally great element A, G or H in ?i 
higher horizontal plane plus the weight of the column of water, 
whose base is this element F and whose height is the distance 
A B = h of: the horizontal planes G H and K L from one another. 
If y is the heaviness of water, this weight is 

G = Fhy, and therefore P, = P + G = P + Fh y. 
If the areas of the elements of surface are unequal ; if, E.G., the 
area of the upper one (in G IT) is F and that of the lower one 
(in K L) is P„ the pressure upon the latter is 




Fx 



F, 



P, - -^ (P + p^ r) = ^ P + P, A y. 



F 



F 



By means of the same formula the pressure P upon an element 



18 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 354. 



F in the horizontal plane G Hc^n be determined, when the exterior 
pressure Pq upon an element of the surface C D =^ F,, which is at 
a distance li above or below G JI, is known. It is 



0? 



F 



F. 



F, ± Fhy, 



Since the pressures upon equal elements in a horizontal plane 
are equal to each other, it follows that the foregoing formula is 
applicable to horizontal surfaces of finite dimension, as, e.g., where 

the water serves to transmit the force P, 
which acts upon a horizontal piston P, 
Fig. 583, to another horizontal piston Fy 
This formula 



Fig. 583. 




P, = ^P + P, h y = P. (y + h y) 



» — 

If we denote the pressures "^ and -^ 



gives directly the pressure Pi upon this 
surface, when h denotes the vertical height 
(7 P between the surfaces of the two pistons. 

Pi 

upon the units of surface by p and^i, we 
have more simply 

Px — p + h y. 



Example. — If the diameters of the two pistons F and Pj of a hydrosta- 
tic press A C B, Fig. 583, are d = 2^ inciies and d^ =9 inches, and if they 
are situated at the distance D = 7i = QO inches above one another, and 
if the larger piston is to exert a pressure i^ = 1600 pounds, we have the 
force which must be applied to the smaller piston 



P = 



F 
F. 



,p,-PA,=y i?--^ 

/ly ..nn - .. 60.63,5 



h 



1600 — T . ^ 



TT 

- u. 
4' ^ 



1728 



= 123,46 — 10,66 = 112,8 pounds. 



§ 354. Surface of Water. — In consequence of the action of 
gravity upon water, all the elements of it tend to descend, and 
really do so when they are not prevented. In order to keep a quan- 
tity of water together, it is necessary to confine it in a vessel. The 
water in a vessel ABC, Fig. 584, can only be in equilibrium when 
the free surface ^ P is at right angles to the direction of gravity, or 



§354.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



719 



Fig. 584. 




Fig. 585. 



horizontal ; for so long as this surface is curved or inclined to the 
horizon there will be elements of the water, such as E, which, be- 
ing situated above the others, will, in consequence of their great 

mobility and their weight, slide down those 
below them as upon an inclined plane. Since, 
when the distances are very great, the direc- 
tions of gravity cannot be considered as paral- 
lel lines, the free surface or the surface of the 
water in a very large vessel, e.g. in a large sea, 
will not, under these circumstances, form a 
plane surface, but a portion of the surface of a 
sphere. If another force acts, in addition to gravity, upon the ele- 
ments of the water, then, when equilibrium exists, the free surface 
of the water is at right angles to the resultant of this force and that 
of gravity. 

If a vessel A B O, Fig. 585, is moved forward with the constant 
acceleration 2^, the free surface of the water forms an inclined plane 

D F\ for in this case every element E 
of this surface is drawn vertically down- 
wards by its weight G and in a horizon- 



tal direction by its inertia P = - 6^, the 

... ^ 

two forces giving rise to a resultant R, 

whose direction forms, with that of 

gravity, a constant angle R E G =^ a. 

This angle is at the same time the angle D F H formed by the 

surface of the water (which is at right angles to the resultant) with 

the horizon. It is determined by the equation 

P p 
tang, a :=.— ■=.'-. 

If, on the contrary, a vessel ABC, Fig. 586, is caused to re- 
volve uniformly about its vertical axis X X, 
the surface of the revolving water forms a 
hollow A C, whose cross-section through 
the axis is a parabola. If w is the angular 
velocity of the vessel and of the water in it, 
G the weight of an element E of the water, 
and y its distance M E from the vertical axis, 
we have the centrifugal force of this ele- 
ment 




Fig. 586. 




-X B 



720 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 354 



F=(s)'^y- (§ 302), 

and therefore for the angle R E G = T E M = (p, formed by the 
resultant with the vertical or by the tangent E Tio the profile of 
the water with the horizontal line M E, 

tang.,t> = ^ = Y- 

From this formula we see that the tangent of the angle, formed 
by the tangent line with the ordinate, is proportional to the ordi- 
nate. Since this is one of the properties of the common parabola 
(see § 157), the vertical cross-section A C oi the surface of the 
water is a parabola, whose axis coincides with the axis of rotation 

XX. 

If the velocity of rotation of the water in the vessel A B D, Fig 



688, were constant and = c, we would have F — 



G 



9 y 



and there 





fore tang. </> = ; hence the subtangent of the curve, formed by 

the cross-section A E B of the water, M T = m = — or constant. 

According to Article 20 of the Introduction to the Calculus, the 
equation of such a curve is 



§355.] EQUILIBRIUM AND PRESSURE OP WATER, ETC. 



721 



,g xc 



-2 



y z=L r e"' = r & 
r denoting the ordinate of the beginning A. 

If we cause a vessel A B H, Fig. 589, to move uniformly in a 
vertical circle around a horizontal axis C, the surface of the water 
will assume a cylindrical form, with a circular cross-section D E H. 
If we prolong the direction of the resultant R of the weight G and 
of the centrifugal force F of an element E until it cuts the vertical 
line C K, passing through the centre of rotation, we obtain the two 
similar triangles ECO and E F R, for which we have 

CO _ F^ _ G^^ 

E G ~ E F ~ F' 

but if we put the radius of gyration E C ^=^ y and retain the last 

(^_Gy 

9 "' 



notations, we have F 



whence it follows that the line 



00 = -S- = ai~-\ 



2936 ^ ^ 894,6 

— 5— leet =■ — -r— meters, 

u u 



Fig. 590. 



u denoting the number of revolutions per minute. Since this value 
of C is the same for all the elements of the water, it follows that 
the resultants of all the elements of the water forming the cross- 
section D E II are directed towards 0, and that the cross-section, 
which is at right-angles to all these directions, is the arc of a circle 
described from 0. Hence the surfaces of the water in the buckets 
of an overshot water-wheel are always cylindrical ones, described 
from the same horizontal axis. 

§ 355. Pressure upon the Bottom. — The pressure in a 
vessel A B C Dy Fig. 590, is a minimum immediately below the 

surface, increases with the depth, and is 
a maximum at the bottom. This, al- 
though a consequence of § 353, can also 
be proved as follows. Let us suppose 
that the area of the surface H^ R^ of the 
water is F^ and that a pressure P„ is ex- 
erted uniformly upon it, e.g. by the at- 
mosphere lying above it or by a piston, 
and let us imagine the entire mass of 
water to be divided by very many hori- 
zontal planes, such as H^ R^, H^ R^, etc., into equally thick layers. 
If F^ is the area of the first layer H, R^, X its thickness, and y the 
heaviness of water, we have the weight of the first layer Gi = F^X y, 
and that portion of the pressure in ITi i?, produced by the pressure 
46 




722 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 335. 



P„ upon the surface of the water H^ E^, according to the principles 
enunciated in § 352, is 

Adding both these pressures, we obtain the pressure in the horizon- 
tal section Hi Ei 

P R 



Pi 



F.. 



Dividing by Fi, we obtain the equation 

P P 

P P 

or, since ~ and -^ denote the pressures 2^^ and p^ in H^ E^ and 

Hx El referred to the unit of surface, we have 

Pi =Po + ^^7- 
The pressure in the following horizontal layer ZTo E^ is deter- 
mined exactly in the same manner as the pressure in the layer 
Hi El, but we must not forget that the initial pressure upon an 
element of the surface is in this case p^ = p^ -\- a y, while in the 
first case it was jJq- Hence the pressure in the horizontal layer 
Hi Eo is 

p, z= p, + Ay = p^ J- A y + A y = ;:>^ + 2 A y, 

Fi®- 591, in like manner the pressure in the third 

layer Hi E^ is 

= 7J, + 3 A y, 

in the fourth 

= ^0 + 4 A y, 

and in the ?^th 

= ;^ + w A y. 
But n X is the depth K = h of this 
^th layer below the surface of the water ; 
we can therefore put the pressure upon each unit of surface in 
the wth horizontal layer 

p — Po -{- li y (compare § 353). 
We call the depth h of one element of surface below the level 
of the water its head or height of water (Fr. charge d'eau ; Ger. 
Druckhohe), and we find the pressure of the water upon any unit 
of surface by adding to the pressure appHed from without the 
weight of a column of water, whose base is unity and whose height 
is the head of water. When a surface is horizontal, as e.g. the 
bottom C D (Fig. 591), the head of water h is the same for all 
positions, and if its area is = F, the pressure of the water upon it is 




§355.J EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



723 



F = (p^ + hy) F= Fp, i- Fhy = F, + Fh y, 
or, if we neglect the external pressure, F — F h y. The pressui^e 
of the water upon a horizontal surface is therefore equal to the tveight 
of the column of water F h above it. 

This pressure of the water upon a horizontal surface, e.g. upon 
the horizontal bottom or upon a horizontal portion of the wall of a 
vessel, is entirely independent of the form of the vessel ; whether 
the vessel A 6', Fig. 592, is prismatic as m a, or wider above than 
below as in h, or wider below than above as in c, or inclined as in 
d, or with spherical walls as in e, etc., the pressure upon the bottom 
IS always equal to the weight of a column of water, whose base is 
the bottom of the vessel and whose height is its depth below the 
level of the water. Since the pressure of water is transmitted in 
all directions, this law is also applicable when the surface, as e.g. 
B C, m Fig. 593, is pressed from below upwards. Each unit of 
surface of the layer of water B K, touching B 0, is subjected to the 
pressure of a column of water, whose height is II B = F F = h, 
and the ]3ressure against the surface C B is = Fh y, F denoting 
the area of that surface. 

Fig. 593. 
A 15 A 



amiiiiiiiiiiiiiiiiiiiiiiiiii^ 





Fig. 594. 




Hence it follows that the water in the communicating tubes 
A B C and DBF, Fig. 594, will stand at the same height, when 
in equilibrium, or that the surfaces A B and F F will be in the 
same horizontal plane. In order to preserve the equilibrium it is 
necessary that the layer of water II R shall be pressed downwards 
by the column of water E R above it as much as it is pressed up- 
wards bv the mass of water below it. Since in both cases the 



724 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 35a 



surface pressed upon is the same, the head of water must be the 
same, and the level of the water at ^ ^ must be at the same height 
above H E as that at U F. 



§ 356. Lateral Pressure. — The formula Just found for the 
pressure of water against a horizontal surface, is not directly appli- 
cable to a 'plane surface inclined to the horizon ; for in this case the 
head of water is different at different points. 

The pressure p = h y upon every unit of surface within the 
horizontal layer at the depth h below the surface of the water acts 
in all directions (§ 352), and, consequently, at right angles to the 
walls of the vessel, by which (§ 138) it ife entirely counteracted. 
Now if F] is the area of an element of the side ABC, Fig. 595, 
and Ai its head of water F^ H^, we have the pressure perpendicular 
^,0 it 

P, = F,.h,y\ 
if F^ is the surface of a second ele- 
ment and 7^2 its head of water, we have 
the normal pressure upon it 

P, = F,h,y', 
and in like manner for a third ele- 
ment 

P, = F^ h^ y, etc. 
These normal pressures form a 
system of parallel forces, whose result- 
ant P is the sum of these pressures, 

p = {F,h, + Fji, + ...)r. 

But i^] ^1 + i^2 ^2 + • • • is the sum of the statical moments of 
F-i, F^, etc., in reference to the surface A B oi the water and 
= Fh, when F denotes the area of the whole surface and h the 
depth S oi its centre of gravity S below the surface of the water ; 
hence the entire normal pressure against the plane surface is 

P = Fhy. 

If we understand by the head of water of a surface the depth of 
its centre of gravity below the surface of the water, the following 
rule will be generally appHcable, viz. : the pressure of 'water against 
a plane surface is equal to the 'weight of a column of water, whose 
base is the surface and whose height is its head of ivater. 

We must here observe that this pressure does not depend upon 
the quantity of water above or in front of the surface pressed, thus, 




§357.] EQUILIBRIUM AND PRESSURE 07 WATER, ETC. 725 

E.G., if the other circumstances are the same, a wall A B CD, Fig. 

596, has to resist the same pressure whether it dams up the water of a 

small trough A C E F or that of a large dam A C G H or that of a 

lake. From the width A B = C I) = b and the height A I) = 

B C = a of the rectangular wall we obtain the surface of the same 

F — ah and the head of 
Fig. 596. ^ 

- H .-"F -^ water SO— - , and, there- 

,^ ^^^ j:f: ^ -"" ^ff ^^^^l^'^^^^B^ fore, the pressure of the wa- 
^''^^ s== ^"^"''^'^ J S1^^ ^^3 ]Mm (-> ter against it is 



^ P — al) . -y = ^^ a^ 1) y. 



~ ^^ ^^*'**&ii*.:*.&;^=^-^ The pressure increases 

therefore with the width and with the square of the height of the 
surface pressed upon. 

Example. — If the water in front of a sluice gate, made of oak, 4 feet 
wide, 5 feet high and 2| inches thick, stands 3^ feet high, how great a force 
is required to lift it ? 

The volume of this gate 

4.5./^== -2/ cubic feet. 
Assuming the heaviness of oak, saturated with water, to be according to 
§ 61, 63,5 . 1,11 = 69,375 pounds, the weight of this gate is 
G = ^- . 69,375 = 25 . 11,5625 = 289,06 pounds. 
The pressure of the water against the gate and the pressure of the lat- 
ter against its guides is 

P = 1 (7)2 . 4 . 62,5 = 49 . 31,25 = 1531,25 pounds ; 
putting the coefficient of friction for wet wood (§ 174) ^ = 0,68, we have 
the friction of the gate upon its guides 

F = ^P = 0,68 . 1531,25 = 1041,25. 
Adding to the latter the weight of the gate, we have the force necessary to 

draw it up 

= 1041,25 + 289,06 = 1330,31 pounds. 

357. Centre of Pressure of Water. — The resultant P = 
Fh y of all the elementary pressures F^ h^ y, F^ h^ y, etc., has, like 
the resultant of any other system of parallel forces, a definite point 
of application, which is called the centre of pressure. By retain- 
ing or supporting this point the whole pressure of the water 
upon a surface will be held in equilibrium. The statical moments 
of the elementary pressures F^ li^ y, F., 1^ y, etc., in reference to the 
plane of the surface of the water A B 0, Fig. 595, are 

F^ A, y . 7^1 = F^ hy" y, Fo lu y . Ih = -^2 Z^/ y, etc., 
and the statical moment of the entire pressure of the water in 
reference to this plane is 



726 



GENERAL PRINCIPLES OF MECHANICS. 



[P57. 



(F, h' + F, K' + . . .) r- 

Denoting the distance K M of the centre M of this pressure from 
the surface of the water by z, we have the moment of the pressure 
of the water 

P z = {F, h, + F^h^ + . , .) z'Y, 

and by putting these moments equal to each other we obtain the 
distance of this centre M below tiie surface of the water 

F, h,' + F, h^ +..._.. F, h;' + F, h,' + 



1).= 



or 



F,h, -^ FJh + ... Fh 

when, as above, F denotes the area of the entire surface and h the 
depth of the centre of gravity below the surface of the water. 

In order to determine completely this point of pressure, we 
must find its distance from another line or plane. If we put the 
distances F^ G^, F^ G-j, etc., of the' elements F^, F^, etc., of the sur- 
face from the line A C, which determines the angle of inclination 

of the plane, = y^, y.^, etc., we have the 
moments of the elementary pressures 
in reference to this line 

— F^ h^ y, y, F. h, y., y, etc., 
and the moment of the entire surface 

= (^1 h, y, 4- F.,h,y, + ...)y', 
denoting the distance M N oi this 
centre M from that line by v, we 
have also this moment 

= {F, h, + F.h, + . . .) V y. 



Equating these two moments, we 
obtain the second ordinate 
2W — -^1 ^^1 .^1 + F.,hy, -\- . . . ^ _ F, h, y, + F., h, y^ + 




F,h, -h F,.h, + ... ^^ Fh 

If a denote the angle of inclination of the plane A B to the 
horizon, x^, x^, etc., the distances F^ F^, Fo F^, etc., of the elements 
i^i, F.2, etc., and u is the distance L M of the centre of pressure M 
from the line of mtersection A B of the plane with the surface of 
the water, we have h^ = x^ sin. a, h^ — x.^ sin. a, etc., and also z = u 
sin. a ; substituting these values of z and v in the expression, we 
obtain 

Moment of inertia 



tt — 



V = 



' + F, X,' + 



F,x, 

F,x^-{-F^x^ + ... 

Fy Xi y^ + Fci x^ y^ -h 

F^ X, + F^Xo -\- . . 



Statical moment 

_ Moment of the centrifugal force 
Statical moment 



358.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



727 



We find then the distances u and v of the centre of pressure 
from the horizontal axis A Y and from the axis A X, formed by 
the hne of dip, when we divide by the statical moment of the sur- 
face with reference to the first axis, in the first place, the moment of 
inertia m reference to the same axis, and, in the second, the mo- 
ment of the centrifugal force of the same in reference to both axes. 
The first distance is also that of the centre of oscillation from the 
line of intersection with the surface of the water. Besides it is 
easy to perceive that the centre of pressure of water coincides per- 
fectly with the centre of percussion, determined in § 313, when the 
line of intersection A Y oi the surface with the surface of the 
water is regarded as the axis of rotation. 

§ 358. Pressure of Water against Rectangles and Tri- 
angles. — If the surface pressed upon is a rectangle A C, Fig. 598, 
with a horizontal base line C D, the centre M of pressure is found 
in the line of dip K L, which bisects the base line, and it is at a 
distance equal to two-thirds of this line from the side A B, which 
lies in the surface of water. If the rectangle, as in Fig. 599, does 

Fig. 598. Fig. 599. Fm. GOO. 






not reach the surface of the water, then, if the distance K L of the 
low^er line C D from the surface of the water = l^ and that K 
of the upper one A B, = l^, we have the distance K M oi the cen- 
tre of pressure from the surface of the water 



II 



t\ Co 



The distance K M of the centre of pressure if of a riglit-angle 
triangle A B C, Fig. 600, whose base A B lies in the surface of the 
water, fi'om A B (Example § 313) is 

''- i F.l ~ ■' ^' 
when / denotes the altitude B C of the triangle. 

The distance of this point M from the other side B C is, since 
this point lies in the line C 0, which bisects the triangle and 



728 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 358. 



runs from the apex C to the middle of the base, JV if = -y = ^ 5, 
h denoting the base A B. 

If the apex C is situated at the surface of the water, as in Fig. 
601, and if the base A B \^ below the apex, we have 

K M — u — -r-^TT = 4 ^ ^^^ 

If the whole triangle ABC, Fig. 602, is immersed in the water, 

and the base ^ i? is at a dis- 



Ftg. 601 



Fig. 603 




U 



tance A H = Ic, and the apex 
C at the distance G H = l^ 
from surface H R, we deter- 
mine the distance M K of the 
centre of pressure M below the 
surface of the water H R ^y^ 
means of the formula 




('. ^ ''^ 



Fig. 603. 



F \l, + 

^ t's (^1 - ^^y + i (^ 4 + i.y ^ k' + 2 /i ^, + 3 4' 

l{%k + k) ~ 2 (?, + 2 k) ' 

The centre of pressure of other plane figures can be determined 
in the same manner. 

Example. — What force Pmust we employ to raise a circular clack- 
valve A B, Fig. 603, which is movable 
about a horizontal axis I) ? Let the 
length of this valve be = 1^ feet, its di- 
ameter J. ^ be = 1;^ feet, and the distance 
of its centre of gravity S from the axis D 
be Z> /S = 0,75 feet, and its weight be 
G^ =z 35 pounds; further, let the distance 
D H of the axis of rotation D from the 
surface of the water, measured in the 
plane of the valve, be = 1 foot and the 
angle of inclination of this plane to thb 
horizon be a = 68°. 

The surface upon which the pressure 
is exerted is 




F = TTr" = 



7:6? 



= 0,7854 . ^ = 1,2273 square feet, 



359.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



729 



and the head of water or depth of its centre G below the water level is 
G =h = H G. sin. a = [HB + D G) sin. a = {H D + BB + B G) sin. a 

= (1 + 0,25 + 0,625) sin. 68° = 1,875 . 0,9272 = 1,7385 feet, 
and, therefore, the pressure of the water upon the surface A B = F\s 

Q = Fhy = 1,2272 . 1,7385 . 62,5 = 133,34 ; 
the arm 5 of this force with reference to the axis of rotation B is the dis- 
tance B if of the centre of pressure ilf from it, hence 

l = HM-HB. 
But we have 

r' . __ 1 



EM = EG ■{■ 



= 1,875 + 



(I)' 



= 1,9271 feet, 



177,95 pounds. 



4: E G ^'"^'^ ' 4. 1,875 
whence 1) = 1,9271 — 1,0000 = 0,9271 feet, 

and the required statical moment of the pressure is 

q'b = 133,34 . 0,9271 = 123,62 foot-pounds. 
The arm of the weight of the valve is 

B K= B8 COS. a = 0,75 . cos. 68° = 0,75 . 0,3746 = 0,2810 feet, 
and therefore its statical moment is 

= 35 . 0,2810 = 9,84 foot-pounds. 
By adding these moments, we obtain the entire moment necessary to 

open the valve 

Pa = 123,62 4- 9,84 = 133,46 foot-pounds. 

Now if the arm of the force, which opens the valve, is B R = a = 0,75 

feet, the intensity of that force must be 

p _ 133,46 

[y - 0,75 

§ 359. Pressure upon Both Sides of a Surface. — If a plane 
surface A B, Fig. 604, is subjected upon both sides to the pressure 

of water, the two resultants of the pres- 
sures on the two sides give rise to a 
new resultant, which, as they act in 
opposite directions, is obtained by sub- 
tracting one from the other. 

If i^" is the area of the portion A B 
subjected to pressure on one side of the 
surface, and li the depth A S of its 
centre of gravity below the surface of 
the water, and if Fi is the area of the portion A^ Bi on the other 
side, which is subjected to the pressure of the water, and Jh the 
depth A^ S^ of its centre of gravity below the corresponding surface 
of the water, the required resultant will be 

P = Fhy - FJi.y = (F h - F, li,) y. 
If the moment of inertia of the first portion of the surface with 
reference to the line, in which the plane of the surface cuts the first 




730 GENERAl. PRINCIPLES OF MECHANICS. [§359. 

surface of the wa^er, = F F, we have the statical moment of the 
pressure of the water upon one side 

= F¥y, 
and if the moment of inertia of the second portion of the surface, 
with reference to its line of intersection with the other surface of 
the water, = F^ hx, we will haye in like manner the statical mo- 
ment of the pressure of the water on the other side, with reference 
to the axis in the second surface of the water, 

= F, /.V y. 

Putting the difference of level A A^oi the two surfaces of the 

water = a, we have the increase of the latter moment, when we pass 

from the axis Ax to the axis A^ 

= F^lix a y, 

and consequently the statical moment of the pressure F^ hi y, in 
reference to the axis A in the first surface of water, is 

= F, lei' y -{■ F,]i,.a.y ^ {F, h,' ^ F, a h,) y. 
Hence it follows that the statical moment of the difference of 
the two resultants is v 

= {F¥ - F, hi' - aFJi,)y, 
and the arm of this difference or the distance of the centre of 
pressure from the axis in the first surface of water is 

_ Fh' ~ Fxh' - aFJh 
^^ ~ ■ FJi- F,1h ' • 

If the portions of surface which are subjected to pressure are 
equal, as is represented in Fig. 605, where the whole surface A B 

= F\^ submerged, we have more simply 
^^^;^^^- P = F{h-h)y, 

and since ¥ = h^^ + '^ a Ih + a" (see § 224) 
:, E , and h — hi = a, we have 
^^"\I^ _ ^'^ ~ ^i' — a hi _ a hi + a" 

_ _ zE^fe. =1 hi + a =^ h. 

^^^^^^^^^^P In the latter case the pressure is equal to 

the weight of a column of water, whose base 
is the surface pressed upon and whose height is the difference of 
level R Hi of the water on the two sides of the surfaces, and the centre 
of pressure coincides with the centre of gra^aty ^S' of the surface. 
This law is also correct when the two surfaces of water are subjected to 
equal pressure, e.g. by means of pistons or by the atmosphere ; for if 
the pressure upon each unit of surface = p and the height of the 

corresponding column of water is Z = - (§ 355), we must substi- 



§360.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 731 



tute. instead of li, li + I, and instead of 7^,, ^1 + /; by subtraction 
we obtain the pressure 

P _-.: (/^ + ^ _ [Aj 4- i\) Fy = {h - h,) Fy. 
For this reason we generally neglect the pressure of the air in 
hydrostatical experiments. 

Example.— The depth A B of the water in the head-bay, Fig. 606, is 
7 feet, the water in the chamber of the lock rises 4 feet upon the gate, and 

the width of the canal and lock-chamber is 
7,5 feet ; what is the resulting pressure upon 
the gate of the lock ? 
Here 

i^ = 7 . 7,5 = 52,5 square feet, 

i^j == 4 . 7,5 == 30,0 square feet, 

1.7, 4 



Fig. 606. 
A 




A=-.7 



2 



» ^1 == o = ^ **6®*> 



2 



6f = 7 - 4 = 3 feet. 



¥ = 



~ and ^1^ 

o 



3 



16 I / / X / ,,, ,, 



hence the required resultant is 

P={Fh- F^ n^) y = (^52,5 . ^ - 30 . 2y 62,5 

= 123,75 . 62,5 = 7734,4 pounds, 
and the depth of the point of application below the surface of the water Is 



52,5. f 



16 
80. --3 



60 



u = 



52,5 



60 



517,5 - 
123/75 



= 4,182 feet. 



§ 360. Pressure in a Given Direction. — In many cases we 
wish to know but one part of the pressure, yiz. : that exerted in a 
certain direction. In order to find such a component, we decom- 
pose the normal pressure M P = P on the surface A B ~ F, Fig. 
607, into two components, one in the given direction J/Xand one 

at right angles to it, viz. : 

M P, = P, andi¥P2 = P,. 
Now if a is the angle P M X 
formed by the direction of the normal 
pressure with the given direction M X 
of the component, the components 
will be 
P^ — P COS. a aud P2 = P sin. a. 
If we project the surface A B upon 
a plane perpendicular to the given di- 
rection M X, we have the area of the projection B C 




732 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 360 



F, = F.cos,ABC, 
or, since the angle of inclination A B G oi the surface to its pro- 
jection is equal to the angle P M X = a, formed by the direction 
of the normal pressure and that of its component Pj, 

Fi — F COS. a, and inversely 

F, 
COS. o.-^\ 

the required component is therefore 



F' 



Now, since the value of the normal pressure is P = i^ ^ y, we 
have P^ — F^li y, 

I.E., the pressure exerted hy water in any direction upon a surface is 
equal to the weight of a column of water, whose hase is the projection 
of the surface at right angles to the given direction and whose height 
is the depth of the centre of gravity of the surface 'below the surface 
of the wafer. 

In most cases in practice we are only required to determine 
the vertical or a horizontal component of the pressure of the 
water against the surface. Since the projection at right angles to 
the vertical direction is the horizontal projection and that at right 
angles to a horizontal direction is a vertical one, we find the ver- 
tical pressure of the water against a surface by treating its hori- 
zontal projection as the surface subjected to pressure, and, on the 
contrary, the horizontal pressure of the water in any direction by 
treating the vertical projection, or elevation, of the surface at right 
angles to the given direction as the surface pressed upon, and in 
both cases we must regard the depth /S' of the centre of gravity 
S of the surface below the surface of the water as the head of water. 
Hence, if we wish to determine in the case of a prisr)iatical em- 
lankment or dam ABBE, Fig. 608, the horizontal pressure of 

the water, we must con- 
sider the longitudinal 
elevation A C, and if 
the vertical pressure is 
to be determined, the ho- 
rizontal projection B C 
of the surface A B must 
be considered as the sur- 
face pressed upon. Put- 




ting 



the length 



of the 



§360.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. >^SS 

dam = I, its height A C = h and horizontal projection of the slope 
B C = a, we have the horizontal pressure of the water 

and its vertical pressure 

Ti 
y — al ,-^y — ^alhy. 

Now if the width of the top of this dam is ^ -E' = 5, the hori- 
zontal projection of the other slope D F =^ a^ and the heaviness of 
the material of the dam = yi, the weight of the dam is 

o = (i + <^-^-^niy„ 

and the entire vertical pressure of the dam upon its horizontal base is 

V+G=ialhy + {h + ^~^)hly, = [lay + (h + ^^)y?^1il 

Putting the coefficient of friction — (p, we have the friction or 
force necessary to push the clam forward 

F:=cp{V+ G) = [iay+{h^- -^) y^] h I 

When the horizontal pressure pushes the embankment forward, 
we must have 

i,h'ly^\lya+ {h + '^^) yj . ^ I, 
or more simply 

7i = /fl^ + (2 J + « + fl^i) ^Y 
If we wish to prevent the dam from being move'd, we must make 

h <,4>\a + {%h + a -^ a,) — j, or 

For the sake of greater security we assume that the water has 
penetrated below the base of the dam to a great extent, and for 
this reason, in the worst case, we must consider that an opposite 
pressure = {h + a + Ux) I h y is acting from below upwards ; 
hence we must put 

A < </> [(2 5 + ^ + «i) ( - - l) - «i} 

Example. — If the density of the clay composing the dam is nearly 
doable that of water, or 

^ = 2 and ^ - 1 = 1, 
7 y 



734 GENERAL PRINCIPLES OF MECHANICS. [§361. 

we can write simply 

h < (f) (2 I) + a). 
It has been found by experiment that a dam resists sufficiently, when 
its height, top and the horizontal projections of its slopes are equal to 
each other. Hence, if we substitute in the last formula 

A = & = », we obtain ^ = |-, 
for which reason in other cases we must put 



h = ^^(2d + a + a,)('^-l^- a,], 



and for clay dams in particular 

h = ^ (2 d + a), OT inversely 

dh — a 

^ = —2—- 

If the height of the dam is 20 feet and the angle of inclination of the 

slope is a = 36°, the horizontal projection is 

a = h cotg. a = 20 . cotg. 36" = 20 . 1,3764 = 27,53 feet, 

and therefore the width of the top of the dam must be 

60-27,53 ,^^, ^ , 

5 = ^—^— = 16,24 feet. 

2 

^7 § 361. Pressure upon Curved Surfaces. — The law of the 

pressure of w^ater in a given direction, deduced in the foregoing 
paragraph, is applicable only to plane surfaces or to a single ele- 
ment of a curved surface, but not to curved surfaces in general. 
The normal pressures upon the different elements of curved sur- 
faces can be decomposed into components parallel to a given direc- 
tion and into others perpendicular to the first. The first set of 
parallel components forms a system of parallel forces, whose result- 
ant gives the pressure in the given direction, and the other set of 
components can also be combined so as to form a single resultant, 
but the two resultants are not capable of further combination, 
unless their directions intersect each other (§ 97). Hence we are 
generally unable to combine all the pressures upon the elements 
of curved surfaces so as to form a single resultant ; there are, how- 
ever, cases where it is possible. 

If Gi, G<i, Gs, etc., are the projections and 7^, Ji., Ih, etc., the 
heads of water of the elements F^, F^, F^, etc., of a curved surface, 
the pressure of the water in the direction perpendicular to the 
plane of projection is 

P = {G, h + G,h, + Gsh, + ...) y, 
and its moment in reference to the plane of the surface of the water is 
P u= { G, Jh' + G, h' + 6^3 h^ -f . . .) y. 

If we can decompose the curved surface subjected to the pressure 



§ 361.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 7a5 



into elements, which have a constant ratio to their projections, 
I.E., if we can put 



=( 



F,h 



0, 


G, 


I'l 


~ F, 


G*. 


F, 


1 


F,Ji, 



= -^ etc. = n, we will have 

^3 



Go = 



?1 
n 



n 



+ 



n 



+ 



...),=( 



etc., and therefore 

F^ ^1 + i^2 ^2 + 



-)r = 



Fh 



n / ' n 

F denoting the area and h the depth of the centre of gravity of the 
entire surface below the level of the water. But we have 
F = F^-\-Fc, +... = nG^ + nG^ +... = n{Gi + G^ + .,.) — n G, 
G denoting the area of the projection of the entire surface ; hence 

y = Ghy, 



P = 



n 




as in the case of a plane surface, or the pressure of water in one 
direction is equal to the weight of a prism of ivater, whose base is 
the projection of the curved surface upon a plane perpendicular to 
the given direction and luhose height is the depth of the centre of 
gravity of the curved surface heloiu the surface of the water. 

Thus, E.G., the vertical pressure against the side of a conical 
vessel A C B, Fig. 609, which is filled with water, is equal to the 

weight of a column of water, whose base is the 
base of the cone and whose height is two-thirds 
the length of the axis CM; for the horizontal 
projections of the surface of a right cone, as 
well as the surface itself, can be decomposed 
into elementary triangles, and the centre of 
gravity S of the surface of the cone is at a dis- 
tance from the apex of the cone equal to two- 
thirds of its height h (§ 116). If r is the radius 
of the base and A the height of the cone, we 
have the pressure upon the base = re r'^ h y and 
the vertical pressure upon the sides = | tt r"" hy \ now as the base 
and the side are united together and the pressures are in opposite 
directions, it follows that the force with which the entire vessel is 
pressed downwards is 

= (1 — I) TT rVi y = ^ TT r'' ^ y 
= the weight of the entire mass of water. If we cut the base loose 
from the conical portion of the vessel it will exert a pressure upon 
its support — TT r"^ h y, and to prevent the side of the vessel from 
being raised by the water we would have to exert a pressure upon 
it = i TT r"^ h y. 



736 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 362. 




Fig. 611. 



Rbmakk. — The pressure exerted by tlie steam of a steam-engine or the 
water of a water-pressure engine is perfectly independent of the shape of 
the piston. No matter how much we may increase tlie 
surface pressed upon by hollowing out or rounding the 
piston, the force, with which the water or steam moves 
the i)iston, remains the same and is equal to the product 
of the cross-section or horizontal projection of the piston 
and the pressure upon the unit of surface. If the piston 
A B, Fig. 610, is funnel-shaped and if its greater radius 
is C A = C B = r and its smaller G D= O E = r^, the 
pressure upon the base is = tt r^ ^ and the reaction 
upon the conical surface is = tc (r^ — T^^)p; hence the 
resulting pressure is 

F = n r"^ p — TT {r^ — r^^) p = irr^^ p 
= the cross-section of the cylinder multiplied by the pressure upon the 
unit of surface. 

§ 362. Horizontal and Vertical Pressure. — Whatever may 
be the form of a curved surface A B, Fig. 611, the horizontal pres- 
sure of the water against it is always equal to the weight of a 

column of water, whose base is 
the vertical projection A^ B^ of 
the surface at right angles to the 
given direction and whose height 
is the depth S of the centre 
of gravity S of this projection 
below the surface of the water. 
The correctness of this assump- 
tion is shown directly by the 
formula 

F = (G,h^ + G,h,+ ...)y, 
when we remember that the heads of water hi, hc^, etc., of the ele- 
ments of the surface are also the heads of water of their projections 
or that Gy hi + Gci hi -\- . . . is the statical moment of the entire 
projection, i.e., the product G h of the vertical projection G multi- 
plied by the depth h of its centre of gravity below the surface of 
the water. Hence we must again put 

F = Ghy 
and remember that h is the head of water of the vertical projection. 
The vertical section, by which we divide a vessel and the water 
contained m it into two equal or unequal parts, is at the same time 
the vertical projection of both parts, the horizontal pressure upon 
one part of the vessel is proportional to its vertical projection 







§362.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



737 



multiplied by the depth of its centre of gravity below the surface 
of the water ; consequently the horizontal pressure upon one portion 
A B of the wall of the vessel is exactly as great as the horizontal 
pressure upon the opposite portion A i B^, which acts in the opposite 
direction, and the two pressures balance each other. The vessel 
will therefore be subjected to equal pressure in all directions by the 
water contained in it. 

The vertical pressure Pi = Gi Jh y of the water against an ele- 
ment F^, Fig. 612, of the wall of the vessel is, since the horizontal 
projection G^ of the element can be regarded as the cross-section, 
Fm. 612. and the head of water hi as the height, or Gi hi 

as the volume, of a nrism, equal to the weight 
of a column of water II Fi, extending above 
the element to the plane H E of the surface 
of the water. Hence the elementary surfaces, 
which form a finite portion A B of the bot- 
tom or wall of the vessel, support a pressure 
which is equal to the weight of the columns 
of water above them, i.e. to the weight of the 
column of water above the entire portion. 
Putting its volume equal to Fj, we obtain 
the vertical pressure of the water 
P = Viy. 
The vertical pressure upon another portion 
Ai Bi of the wall of the vessel, which lies 
vertically above the former and which limits the volume Ai BiH = 
Fs, is 

Q= V,y; 
but if the two portions are rigidly connected together, the result- 
ant of the two forces, which acts vertically downwards, will be 

E= {F - Q) = {Vi - FO y=Vy 
= to the iveight of the column of ivater contained between the two 
surfaces. If we apply this rule to the entire vessel, it follows that 
the entire vertical pressure of the water against the vessel is equal to 
the weight of the water contained in it. 

If we make an openmg in the side of the vessel H B R, Fig. 
613, I and II, that portion of the pressure, which corresponds to 
the cross-section of this opening, is wanting and the pressure upon 
the surface F opposite to it remains unbalanced. If the opening 
(as in I) is closed by a stopper K, which is prevented from yielding 
by a resisting object L on the outside, an equal distribution of the 
47 




738 



GENERAL PRINCIPLES OF MECHANICS. 



[§ B63. 



horizontal pressure upon the walls of the yessel no longer takes 
place, but, on the contrary, the vessel is moved forward with a 
force P = Fhy, which is counteracted on the opposite side by 

Fig. 613. 

I n 




Fig. 614. 



the stopper. If the stopper is removed and the water allowed to 
flow through the opening 0, as m II, the reaction of the discharg- 
ing water increases this pressure P from F h y to Pi — % F li y, 
as will be shown hereafter. 

Example. — The vertical pressure Pj upon the lower hemispherical sur- 
face ABB, Fig. 614, is equal to the weight of a column of water bounded 

above by the surface of the water H B and below 
by the hemispherical surface. If r is the radius C A 
= C B o'l this surface and Ti the height G of the 
surface of the water above the horizontal plane A B, 
wlii'jh limits :t, the volume of the hemisiDhere ABB 
will be Fj = f TT r^, and that of the cylinder above 
A B, V^ = n r- h; hence 

The pressure, which is directed vertically upwards 
upon the upper hemisphere ABB, is, on the contrary, 

and tjaerefore the entire vertical pressure 

is equal to the weight of water in the entire sphere. 

The horizontal pressure upon one of the hemi- 
spheres B A E and B B E, which join each other in 
the vertical plane B C E, is measured by the weight of a prism, whose 
base \s B C E = irr^ and whose height is := h-, this pressure is 
jb{ B = 7T r"^ h y. 

§ 363. Thickness of Pipes. — The appHcation of the theory 
of the pressure of water to the determination of the thickness of 
pipes, boilers, etc., is of great importance. In order that these 
vessels shall sufficiently resist the pressure of the water and not be 




363.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 



739 



broken, their tvalls must be made of a certain tliickness, which de- 
pends upon the head of water and the internal diameter of the 
vessel. The rupture of the pipe may be caused either by a trans- 
verse or by a longitudinal tearing. The latter form of rupture is 
most likely to occur, as will appear from the following discussion. 

If the head of the water in a pipe = li or the pressure upon the 
unit of surface of the pipe is 7; = li y, the width of the pipe M N 
= 2 C M = 2 Vy Fig. 615, and the cross-section of the body of 
water in it i^ = n r^, the pressure, which is exerted upon the sur- 
face of the end of the pipe and which must be sustained by the 
cross-section of the tube, is 

P — F p = 7r rVi y = TT r'^ p. 

Now if the thickness of the pipe \^ A D — B E = e, its cross- 
section is 

= 7T(r + e)'' — Trr' = 27rre + e^ = 27rre(l + ^J, 

and if we denote the modulus of proof strength of the material, of 
which the pipe is composed, by T, the proof strength of the entire 
tube in the direction of the axis is 

Hence we can put 

(1 + —J 2 7T r e T = n r'^ p, or 

(1 +^)2«r=rp(see§205); 

the resolution of this equation gives 
the thickness 

r p 




e = 



2 



(' - i^ 



T 



of the pipe, for which we can generally write with sufficient accu- 
racy 

_ r p _ rhy 

^ ~ Ylr '~ 2~T' 
The mean pressure, which the water exerts upon a portion of 
the wall A M B, whose length is I and whose central angle is 
A C B — 2 a'', is, since the projection of this portion at right 
angles to the line C M passing through the centre is a rectangle, 
whose area \^ A B .1 = 2 r I sin. a, 

P — 2rl sin. a . p = 2 r I h sin. a . y. 



740 GENERAL PRINCIPLES OF MECHANICS. [§363. 

This force is held, in equilibrium by the forces of cohesion i?, 
R in the cross-sections A D .1 and B U . I = e I of the wall of the 
pipe ; it is therefore equal to the sum 2 Q of the components 
B Q = Q and. B Q = Q oi the latter forces, which are parallel to 
the line C M. Now if we put R — el T,^^ obtain 

Q — R sin, A R Q — R sin. A C M = e I T sin, c, 
and therefore 

2 e I T sin. a — % r I p sin. a, i.e. e T =^ r pi 
hence the required thickness of the pipe is 

_r p _ rliy 

which is entirely independent of its length. 

Since the first calculation gave e only = "977? I't follows that 

to prevent a longitudinal tear we must make the wall twice as 

thick as would be necessary to prevent a transverse one. 

iFrom the formula 

_ r p _ rhy 
^^^ - -^r- 

just found, it follows that the thickness of similar pipes must he 
proportional to the width and to the head of ivater or pressicre upon 
the unit of surface. A pipe, which is three times as wide as 
another and which has to bear a pressure five times as great as the 
first, must be fifteen times as thick. 

We must give to holloiv spheres which sustain a pressure p upon 
each unit of surface the thickness 

r p 

for here the projection of the surface pressed upon is the great 
circle t r^, and the surface of separation of the ring is 2 tt r e 

(1 + ^— J, or approximatively, when the thickness is small, — 2 TT re. 

The formulas just found, give for jo = also e = ; hence 
pipes, which have no internal pressure to resist, can be made infi- 
nitely thin ; but since every pipe in consequence of its own weight 
must sustain a certain pressure and also must be made of a certain 
thickness to be water-tight, we must add to the value found a 
certain thickness e in order to have a pipe, which under all circum- 
stances will be strong enough. Hence for a cylindrical tube or 
boiler we have 



§363.] EQUILIBRIUM AND PRESSURE OF WATER, ETC. 741 



or more simply, if d is the interior width of the tube, ^ the pressure 
m atmospheres, each corresponding to a column of water 34 feet 
high, and ft a coefScient determined by experiment, 

e = 61 + fji p d. 
It hag been experimentally determined that for tubes made of 



Sheet iron 
Cast iron 
Copper . 
Lead 
Zinc , . 
Wood , 
Natural stone 
Artificial stone 



a 



<( 



a 



a 



a 



a 



a 



e = 0,0008'6 p d + 0,12 inches, 
e = 0,00238 p d -h 0,34 
e = 0,00148 j9 d + 0,16 
e = 0,00507 pd + 0,21 
e = 0,0024.2 pd + 0,16 
e = 0,0323 pd -h 1,07 
e = 0,0369 p d + 1,18 
e = 0,0538 p d -i- 1,58 
Example.— If a vertical water-pressure engine has an inlet cast-iron 
pipe 10 inches wide inside, how thick must its walls be for a depth of 100, 
200 and 300 feet ? For a depth of 100 feet tliis thickness is 0,00238 . 
-^^ . 10 + 0,34 = 0,07 + 0,34 = 0,41 inches ; for a depth of 200 feet = 
0,14 + 0,34 = 0,48 inches; and for a depth of 300 feet = 0,21 + 0,34 = 
0,55 inches. Cast-iron conduit pipes are generally tested to 10 atmo- 
spheres, in which case we have 

e = 0,0238 d + 0,34 inches, 
and for pipes of 10 inches internal diameter we must make the thickness 

e = 0,24 + 0,34 = 0,58 inches. 

Remark — 1) In the second part of this work the thickness of tubes ex- 
posed not only to hydrostatic pressure, but also to hydraulic impact, will 
be calculated. 

2) In the second part the thickness of the walls of steam-boilers will be 
treated. Upon the theory of the thickness of pipes, we can consult the 
treatise of Geh. Regierungsrath Brix in the proceedings of the " Vereins 
zur Beforderung des Gewerbefleisses, in Preussen," year 1834, and Wiebes 
'' Lehre von den einfachen Maschinehtheilen," Vol. I, and also Rankine's 
" Manual of Applied Mechanics," page 289, and Scheffler's " Monographien 
iiber die Gitter- und Bogentriiger, und liber die Festigkeit der Gefasswdnde.'''' 
The technical relations and the testing of pipes are treated in Hagen's 
" Handbuch der Wasserbaukunst," Part 1st, and also in Geniey's "■ Essai 
sur les moyens de conduire, etc., les eaux," and in the " Traite theoretique 
et pratique de la conduite et de la distribution des eaux," par Dupuit, 
Paris. 1854. 



742 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 364. 



CHAPTER II. 

EQUILIBRIUM OF WATER WITH OTHER BODIES. 

§ 364. Upward Pressure, Buoyant Effort. — A body m- 
mersed in neater is subjected to pressure upon all sides, and the 
question arises, what is the magnitude, direction, and point of 
application of the resultant of all these pressures ? Let us imagine 
this resultant composed of a yertical and two horizontal compo- 
nents, and let us determine them according to the rules of § 362. 
The horizontal pressure of the water against a body is equal to the 
horizontal pressure against its vertical projection ; but every eleva- 
tion A C, Fig. 616, of a body is at the same time the projection of 
the rear part ADC and of the fore part A B C oi its surface, and 
consequently the pressure P upon the hind part of the suiface of a 
body is equal to the pressure — P upon the fore part ; and as the 
directions of these pressures are opposite, their resultant is = 0. 
Since this relation exists for any given horizontal direction and its 
corresponding vertical projection, it follows that the resultant of 
all the horizontal pressures is equal to zero, and that the body A C, 
which is under water, is subjected to equal pressure in all horizontal 
directions^ and therefore has no tendency to move horizontally. 



Fig. 616. 





y Ymmmiii^mi>f<9, "%}mm(mmri%u. 



In order to find the vertical pressure of the water upon a body 
A B D, Fig. 617, immersed in it, let us imagine it to be decomposed 



§365.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 743 

into the yertical elementary prisms A B, CD, etc., and let us de- 
termine the vertical pressure upon their bases A and B, C and D, 
etc. Let the lengths of these columns be l^, 4, etc., the depths 
H B, K D of their upper ends B, D below the surface of the water 
R\)Q Ih, Ih, etc., and their horizontal cross-sections be F^, F.2, etc., 
then we have the vertical pressures which act from above down- 
wards upon their ends B, D, etc., 

Q„ Q„ etc., = Fy Ji, y, F^ h^ y, etc., 
and, on the contrary, the vertical pressures which act from below 
upwards against the ends A, C, etc., are 

P„ Po, etc., = F, {Ih + I,) y, F^ {h, + 4) y, etc. 
By combining these parallel forces we obtain the resultant 

P = Pi + P2 + ... - (^1 + ft + ...) 

= Pi {hi + li)y + F, (7h + k) y + . . . - FJi, y - F,h,y- , . . 

= (F,l, + P, /, + ...)y =:Vy, 

in which V denotes the volume of the immersed body or of the 
w^ater displaced by it. Hence the iipivard pressure or buoyant effort, 
^oith ivhich luater tends to raise up a hody immersed in it, is equal to 
the weight of the water displaced or of a quantity of luater tvhich has 
the same vohcme as the stibmerged body. 

Finally, in order to determine the point of application of this 
resultant, let us put the distances E F^, E P^, etc., of the elemen- 
tary columns A B, G D, etc., from a vertical plane X equal to 
a^, a^, etc., and let us determine their moments in reference to this 
plane. If S is the point of application of the upward thrust, which 
is called the centre of buoyancy, and E S — x its* distance from 
that plane, we have 

F y :?: — Pi /i y . «i + P2 4 7 • ^2 + . . ., 
and therefore 

__ F^l^a^ + P2 4 «2 + • • • _ ViCf'A + Fo <X2 + • . . 
^ ~ P, k-\- FJ, + ... ~ Fj +F2 + ... ' 
Fi, F2, etc., denoting the contents of the elementary columns. Since 
(according to § 105) the centre of gravity of a body is determined 
by exactly the same formula, it folloics that the point of application 
S of the upward thrust coincides with the centre of gravity of the 
water displaced. The direction of the buoyant effort is called the 
line of support ; when it passes through the centre of gravity of the 
body, it is called the line of rest. 

§ 365. Upward Pressure, or Buoyant Effort, when the 
Body is Partially Surrounded by Water. — If a body, such as 

A B D, Fig. 618, is not entirely surrounded by the water A II R, 



744 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 365, 



and the surface A B, whose area is F, is united to the wall of the 
yessel, or if the body, where its cross-section is ^ B = F, passes 
through the wall of the vessel, the pressure which the water would 
have exerted upon this surface A B, if the body was free or in con- 
tact with the water alone, is absent. 

If we denote the head of water upon 
A B, I.E. the depth of its centre of grav- 
ity below the surface of water H R, by 
h, the pressure of the water upon A B 
will be P = F h y\ and if Vi denotes 
the volume of water displaced hj A BD^ 
the buoyant effort of the water, or the 
force, with which the body would tend 
to rise if it were free, is Pi = Fi y. 

However, since the pressure upon 
^ P is wanting, the entire action of the 
water upon the body is the resultant P 
of Pi = Fi y and - P = - P h y. 
In order to determine this resultant, we prolong the vertical line 
of gravity of the water displaced and the right line passing through 
the centre M of the pressure perpendicular to A B until they meet 
at the point C; then, assuming the forces Pi and — P to be ap- 
plied at this point, we combine them by means of the parallelogram 
of forces and obtain the resultant C E = E. 

If the inclination of the surface ^ P to the horizon as well as 
the deviation of the force P from the vertical = a, the angle 
formed by the directions of the forces P and — Pi with each other 
will be = if (7 ii = 180 — a, and therefore the resultant, which 
measures the whole effect of the pressure of the water upon the 
body ABB, will be 




E= VIT + P' -% P P, 



COS. a 



= yVV^' + {Fhy -- 2 V,Fh COS. a. 

According to the principle of action and reaction, the body will 
react with a pressure — E upon the water. If F^ is the volume 
of the water in the vessel or V^ y its weight G, the pressure, which 
acts vertically downwards upon the vessel, is 

Q= T^o r + ^1 = (F„ + Fi) y, I.E. Q = Vy, 
when F = F^ + F denotes the volume of the space occupied by 
the water and the body A B D. 

Combining this with the pressure P = P ^ y, we have the entire 
pressure sustained by the vessel 



§366.] EQUILIBrAjM OF WATER WITH OTHER BODIES. 745 



R^ = VQ' + F' -2 QF COS. a 



Fig. 619. 



= yVV' + {Fhy -2VFh cos, a. 
If the surface A B were horizontal or a == 0", we would haye 

R= {V,- Fh)y^ndiR, = {V - F h) y. 
If also Fi = 0, R would be = -Fhy (see § 355). 

§ 366. Equilibrium of Floating Bodies. — The buoyant 
effort F upon a body floating or immersed in water is accompanied 
by the weight G of the body, which acts in the opposite direction, 
and the resultant of the two forces is 

R= G - FoY^ {e -l)Vy, 

in which e denotes the specific grayity of the body. 

If the body is homogeneous, its centre of gravity and that of the 
water displaced coincide, and this point is consequently the point 
of application of the resultant R ^= G — F\ but if the body is 
heterogeneous, the two centres of gravity do not coincide and the 
point of application of the resultant does not coincide with either 
of them. Putting the horizontal distance 8 H, Fig. 619, of the two 

centres of gravity from each other = b 
and the horizontal distance /S^ A of the 
required point of application A from the 
centre of gravity S of the water dis- 
placed, = a, we have the equation 

Gb = Ra, 
whence we obtain 

_G^_ Gb 
^ ~ R ~ G - P 
If the immersed body is abandoned to 
the action of gravity, one of three cases 
may occur. Either the specific gravity 
e of the body is equal to that of the water, or it is greater, or it is 
less. In the first case the buoyant effort is equal to the weight, in 
the second it is smaller, and in the third it is greater. While in 
the first case the buoyant effort and the weight are in equilibrium, 
in the second case the body will sink with the force 

a -Vy = {e -\) Vy, 
and in the third case it will rise with the force 

Vy - G ^ {\ - e) Vy. 
The body will continue to rise until the volume V^ of the water 
displaced by the body and limited by the plane of the surface of the 




746 



GENERAL PRINCIPLES OF MECHANICS. 



[p67 



Fig. 620. 




water has the same weight as the entire body. The weight G = 
F e y of the body A B, Fig. 620, and the buoyant effort P — 

Vi y form a couple, by which the body is turned 
until the directions of these forces coincide or 
until the centre of gravity of the body and the 
centre of buoyancy come into the same vertical 
line, or until the line of support becomes a line 
of rest. From the equality of the forces P and 
G we have the expression 

F, = e r, 01- -p = J. 

The line passing through the centre of gravity of the floating 
body and the centre of buoyancy is called the axis of floatation (Fr. 
axe de flottaison ; Ger. Schwimmaxe), and the section of the float- 
ing body formed by the plane of the surface of the water is called 
the plane of floatation (Fr. plan de flottaison; Ger. Schwimme- 
bene). From what precedes we see that any plane, which divides 
the body in such a manner that the centres of gravity of the two 
portions will be in a Ime perpendicular to it, and that one portion 
of the body will be to the whole as the specific gravity of the body 
is to that of the liquid, will be a plane of floatation of the body. 

§ 367. Depth of Floatation, — If we know the form and 
weight of a floating body, we can calculate beforehand by the aid 
of the foregoing rule the depth of immersio7i. If G is the weight 

of the body, we can put the volume of the 

water displaced 

y 

if we combine this with the stereometric formula 
for this volume Fj, we obtain the required 
equation of condition. 

For a prism ABC, Fig. 621, whose axis is 
vertical, we have V^ = F y, when F denotes the 
cross-section and y the depth C D of immer- 
sion ; hence it follows that 

T, G ^ G G h 

F y — — and y = -=— = ~= — , 

^ y ^ F y V y 

in which F denotes the volume and li the length of the fl oating 
prism. 

For a pyramid ABC, Fig. 622, floating with its apex below 



* 


Fig. 


621. 






A E 


B 




H 




III 


1 




R 




=-.Hi...i;^ ,, 


ilillliltF=^ 


■ 1 __ 




==:;!i.!miiiiiiii 






»^- 




— -=^^_— 


.=- 




'-in:::: - 


^^^- 


--^^ 


; !„, 


:;;:ni;!,;ii ~ 





^" 


si:iiiiiiiil|ll 


:i^Si^= 


— 


— ^= 


Esriiiaisjjijgi^ 




— = 


--- 






llWilSEl 




^^ — 






=— 


j:=- 


A'r 







:r~ 



§367.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 747 

the surface of the water, we have, since the contents of similar 
pyramids are proportional to the cubes of their heights, 

V ~ ~F' ^^^ consequently the depth of imf}iersion, is 

in which F denotes the volume and ^ the height of the pyramid. 
Fig. 622. 
A E B 





For a pyramid, ABC, Fig. 623, floating with its base under 
ivater, we obtain, on the contrary, the distance C D — y^ from the 
apex to the surface of the water by putting 

F ~ 



3 r 



/ V ^ / r^ 

, whence y, = h f 1 ~ i ^hi/l- ~. 

1/ y y 



h' ' "' - r - y 

For a sphere A B, Fig. 624, whose radius i^ G A = r, 

y 



V,=ny'(^r- |], 



Fig. 634. 



we have therefore, in this case, to 
solve the cubic equation 

n y 



in order to find the depth of the 
immersion i> ^ = ^ of the 
sphere. 

If a cylinder A K, Fig. 625, 

floats with its axis horizontal and 

its radius is ^ C = B C = r,WQ have, when a" denotes the central 

angle A C B oi the immersed arc, for the depth of immersioh D E 

y = r (1 — COS. ^ a) ; 
now in order to find the arc a we must put the volume of the water 




748 GENERAL PRINCIPLES OF MECHANICS. [§368: 

displaced = sector {^j minus the triangle ^ 2/' "^^^*^P^^®^ 

by the length B K — I of the cylinder, or 

Fig. 635. \^ ~ ^^^' ^^ 2 "~ Y' 

jljl^^^^^^ and resolve the equation 



a — sm. a = 



Ir^ y 
^^l|iiipji^^^^ ^ K:^=^^^=^ by approximation with reference 

^—=^^=^^^^^ ^ ==r3S3=^=s^,:^^^ Example — 1) If a wooden sphere 

10 inches in diameter, which is float- 
ing, is immersed 4| inches in the water, the volume of the water displaced is 

Fi = TT (f )^ (5 - I) = ""• • - = — g^ = 332,66 cubic inches, 

while tbe volume of the sphere itself is 

= 533,6 cubic inches. 



6 6 

Therefore 533,6 cubic inches of the material of the sphere weigh as much 

as 333,66 cubic inches of water, and the specific gravity of the former is 

333,66 

' = -533;^ = ^'^^^- 

3) How deep will a wooden cylinder 10 inches in diameter sink, when 

floating, if its specific gravity is e = 0,435 ? Here 

a — sin. a tt r^ I . e y ^ .^^ ^ ^^..^ 

= -^r-^ = 7r £ = 0,435 . n = 1,3353. 

3 I r^ y 

Now the table of segments in the " Ingenieur," page 154, gives for the area 

a — sin. a 

= 1,33766 a segment of a circle, whose central angle is a° = 

ft — HOT} n 

166°, and for — ^ — = 1,34487 an angle a° = 167" ; we can, therefore, 

put the angle at the centre, corresponding to the sector 1,3353 

,0 _ i66„ ^ 1,33530 - 1,33766 _ ^ 754^ _ 

"" - ^'^^ ^ 1,34487-1,33766 • 1 - 1^^ + 1731 - ^^^ ^^ • 
The depth of immersion is, therefore, 
2/ = r (1 — COS. 1 a) = 5 (1 — COS. 83" 13^ = 5 . 0,8819 = 4,41 inches. 

§ 368. The most important application of the above principle 
is to the determination of the depth of immersion of boats and 
ships. If the boats have a regular form this depth can be calcu- 
lated by geometrical formulas ; but if the form is irregular, or if its 
equation is unknown, or if it is composed of very many forms, the 
depth of immersion must be determined by experiment. 



368.] EQUILIBRIU'M OF WATER WITH OTHER BODIES. 749 



An example of the first case is furnislied by the boat AC E QH, 
represented in Fig. 626, whose sides are plane surfaces. It con- 



FiG. 626. 




sists of a parallelopipedon A G F and two four-sided pyramids 

C E F and B G H, which form the bow and stern, and its plane 

of floatation is composed of a parallelogram K L P and of two 

trapezoids L M N and K P Q R which limit the space, from 

which the water is displaced and which can be decomposed into a 

parallelopipedon K G T, into two triangular prisms U V M N 

and W X R Q, and into two four-sided pyramids G F if and B X R. 

Let us put the length A D = B G oi the central portion = I, its 

width A G — h and its height A B = h, the length of each of the 

two beaks = c and the depth of immersion under water, i.e. B K 

= G L = y. It follows that the immersed portion K G T oi 

the middle piece is 

= WG.'G^.ITL = lly. 

Putting the width G U of the base of the pyramid G V M, = x 

and the height of this pyramid =: z, we have 

X z y ^ 

T — - = T> whence 

c h 

h , c 

hence the volume of this pyramid is 

and those of the two pyramids (C V M and B X R) together are 






The cross-section of the triangular pyramid TJ V N \^ 

and the side Jfi\r= V 



1 ^ y 



^h-ly^ 



= j 



h 



= ^(^-tl-' 



750 GENERAL PRINCIPLES OF MECHANICS. [§369^. 

hence the contents of the two prisms V U N and X W Q together 
are 

Finally, by adding the volumes first found, we obtain that of 
the water displaced 

Now if the gross weight of the boat = G, we must put 
(^ + ^^- - i . f-) ^ ^ r = 6^, or 

y^ — ^ h y — . y + -, — - = 0. 

^ ^ c c y 

By resolving this cubic equation we obtain from the gross 

weight G of the boat its de2Jth y of floatation. 

Example — 1) If the length of the middle portion of a boat is Z = 50 
feet, the length of each terminal pyramid is c = 15 feet, the width & = 13 
feet and the depth A = 4 feet, the total load for an immersion of 2 feet is 

(? = [50 + 15 . 1 - 1 . 15 . (1)^] .12.2. 62,5 
= [50 + 7,5 — 1,25] 24 . 62,5 = 84375 pounds. 

2) If the gross weight of the above boat was 50000 pounds, we would 
have for the depth of immersion 

2^3 _ 12 ^2 _ 160 ^ ^ 213,33 = 0. 
From this we obtain 

213,33 + 2/' — 12 2/' 
y = ^:^ = 1,333 + 0,00625 y' - 0,075 y\ 

approximatively, y = 1,333 + 0,00625 (1,333)^ — 0,075 (1,333)=* 
= 1,333 + 0,0148 — 0,1333 = 1,215, and more exactly 
y = 1,333 + 0,00625 (1,215)^ - 0,075 (1,215)^ = 1,2338 feet. 

Remark. — In order to find the weight of the cargo, vessels are provided 
on both sides with a scale. The divisions of the scale are generally deter- 
mined empirically by finding the immersion for given loads. This subject 
will be treated more at length in the third volume. 

§ 369. Stability of Floating Bodies.— A body floats either 
in an upright or inclined position, and tvith or without stability. 
A body, E.G. a ship, floats in an upright position, when at least one 
of tha-^ planes passing through the axis of floatation is a plane 
of symmetry of the body, and in an inclined position, when the 
body cannot be divided into two symmetrical parts by any plane 
passing through the axis of floatation. A floating body is in stable 



369.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 



751 



equilibrium, when it tends to maintain its position of equilibrium 
(compare § 141), i.e. if work must be done to move it out of this 
position, or if it returns to its original position of equilibrium aftei* 
having been moved from it. A body floats in unstable equilibrium, 
when it passes into a new position of equilibrium as soon as it has 
been moved from its original one by being shaken, by a blow, etc. 

If a body A B C\ Fig. 627, which was floating in an upright 
position, is brought into an inclined one, the centre of buoyancy 8 
moves from the plane of symmetry and assumes a position Si in 
the half of the body most immersed. The buoyant effort P = V y, 
which is applied at Sx, and the weight of the ship G = — P, which 
is applied at C, form a couple which will always turn the body 
(see § 93). No matter around what point this rotation takes place, 
the point C, yielding to the weight G, will always sink, and the 
point 8i or another i¥, situated in the vertical line 8i P, yielding 
to the action P, will rise, and the axis or plane of symmetry E F 
will be drawn downwards at C and upwards at M, and therefore 
the body will right itself when M, as in Fig. 627, is above C, and. 



Fig. 627 

A 





on the contrary, it will incline itself more and more when, as is 
represented in Fig. 628, M is situated below C. Hence the stability 
of a floating body, such as a ship, depends upon the point M, where 
the vertical line, which passes through the centre of buoyancy 8^, 
cuts the plane of symmetry. This point is called the metacentre 
(Fr. metacentre; Ger. Metacentrum). A ship or any other body 
floats with stability when its metacentre lies above its centre of 
gravity, and without stability when it lies below it ; it is in indif- 
ferent equilibrium when these two points coincide. 

The horizontal distance CD of the metacentre M from the 
centre of gravity C of the ship is the arm of the couple formed by 
P and G — — P, and its moment, which is the measure of the 



752 GENERAL PRINCIPLES OF MECHANICS [§370. 



stability, is — P . G D. If we denote the distance C M by c, and 

the angle S M Si, through which the ship rolls or through which 

its axis is turned, by </>, we have for the measure of the stability of 

the ship 

S = P c sin. ; 

it increases, therefore, with the weight, with the distance of the 
metacentre from the centre of gravity of the ship and with the 
angle of inclination. 

§ 370 Determination of the Moment of Stability. — In 

the last formula 

S = P c sin. 0, 

the stability of the ship depends principally upon the distance of 
the metacentre from the centre of gravity of the ship, and it is, 
therefore, important to obtain a formula for the determination of 
this distance. While the ship ABE, Fig. 629, passes from its 

upright to its inclined position. 

Fig. 639 ^|^g centre of buoyancy 8 moves to 

^'^ Si, and the wedge-shaped space 



H Hi passes out of the water 
drawing the wedge-shaped piece 
R Ri into it, and the buoyant 



^^ ' '','^if^ drawing the wedge-shaped piece 




^-- !S=^" effort on one side is diminished 

r /^g^ by the force Q, acting at the cen- 

m^d-^mi tre of gravity i*^ of the space 
H Hi and upon the other side 
it is increased by an equal force Q, 
acting at the centre of gravity O 
of the space R R^. Therefore 
the force P applied at Si replaces 
the force originally applied at S and the couple ( Q, — Q), or, what 
amounts to the same thing, an opposite force — P, acting in Si, 
balances the force P applied at S together with the couple {Q, — Q), 
or more simply a couple (P, — P), whose points of application are 
S and Si, balances the couple {Q, — Q). ]^ow if the cross-section 
H E R = Hi E Ri of the immersed portion of the ship = F and 
the cross-section H Hi = R Rioi the space, which is drawn 
out the water on one side and immersed on the other, = Pj, if the 
horizontal distance K L of the centres of gravity of these spaces 
from each other = a and the horizontal distance M T of the centres 
of gravity S and Si from each other, or the horizontal projection 



§370.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 753 

S Si of the space described by ^S^, during the rolling, = 5, we have, 
since the couples balance each other, 

F s = F-^ a, whence s = -^ a and 

M T s F^a 



8M = 



sin. (f) sin. 4> F sin. (f 

The line C M = c, which enters as a factor into the measure 
of the stability, h = G S -{■ 8 M\ denoting, therefore, the distance 
C 8 oi the centre of gravity (7 of the ship from the centre of buoy- 
ancy 8 by e, we obtain the measure of the stability 

8 — P c sin. (j) = P l—^ — h e sin. 0). 

If the angle through which the ship rolls is small, the cross- 
sections H Hi and R Rx can be treated as isosceles triangles. 
If we denote tlie width H R = Hi Ri oi the ship at the surface of 
the water by h, we can put 

as well as sin. = 0; hence the measure of the stability of the 
ship is 

If the centre of gravity G of the ship coincides with the centre 
of buoyancy 8, we have e = 0, whence 

and if the centre of gravity of the ship lies above the centre of 
buoyancy, e, on the contrary, is negative and 



^ = (i&-'')^^- 



It also follows that the stability of a ship becomes null, when e 

IS negative and at the same time = „ . 

We see from the results obtained that a ship^s stability is greater 
the wider the ship is and the lower the centre of gravity is. 

Example. — The measure of the stability of a parallelopipedon A D, 

Fig. 630, whose width A B = h, whose height A E = h and whose depth 

7i — y 
of immersion E H = y is, since F = h y and e = — —^ — , 

48 



754 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 371. 



or if the specific gravity of tlie material of which the parallelopipedon is 
composed be put = e 

From this we see that the stability ceases 
when 

d^ = 6h^ e (1 — e), I.E., when 



Fig. 630. 




©^ 



y = V6 £ (1 
For f = 1^ we have 



0- 






Vt 



1 

"2" 



V| = 1,225. 



If in this case the width is not at least 1,225 times the height, the paral- 
lelopipedon floats in unstable equilibrium. 

371. Inclined Floating. — The formula 

'F^ a 



S 



F j -^^ ± e sin. I 



for the stability of a floating body can also be employed to determine 
the various positions of floating bodies ; for if we put S = 0, we 
obtain the equation of condition of the position of equilibrium, and 
by resolving it Ave obtain the corresponding angle of inclination. 
We have, therefore, to resolve the equation 

F, a 



F 



± e sin. = 



Fig. 631. 
B 



in reference to (p. 

In the case of a parallelopipedon A B D E, Fig. 631, the cross- 
section i^is = HRDE—H^RxBE— i y,i denoting the 
width A B = H R and y the depth of immersion F H = D Ry 

and the cross-section 

F, = HOH,^-ROR, 
is a right-angled triangle, whose base 
is H ^ R --^ ^,1, 
and whose altitude is 

H H, = R Ri = J 5 tang. <f), 
whence 

F, = ^y tang. (p. 



Now since the centre of gravity P 
is at a distance 

FV=IHH, = Ihtang.^ 




§371.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 755 

from the base H R and at a distance U = % II — I b from 
the centre 0, it follows that the horizontal distance of the centre 
of gravity F from the centre 

= K = ^^ + N K = Ucos. (p + F U sin. 

= 3 1) COS. <P + ^ i tang, (f) sin. 0, 
and the arm of the lever is 

a = TL = 2 OK =ih COS. (p + i b --^. ' 

"^ ^ COS. (I) 

Hence the equation of condition of the inclined position of 

equilibrium is 

X If tana. (j> (% h cos."" 6 -l- I b sin.^ 0) • . /^ 

b y COS. (p 

sin (f) 
or, substituting — '—- = tang, (b, 
' ^ cos.d) ^ 

sin. [(jo + ^V tangf (})) b^ — e y] = 0, 

which equation is satisfied by 

sin. = and by 

tang. = V2 V^^ - 1. 

The angle -— 0, determined by the first equation, is applicable 
to the body when in an upright position, and that, given by the 
second equation, to the body when floating in an inclined position. 

If the latter case is possible, yf must be > -^\. Now if h is the 

height and e the specific gravity of the parallelopipedon, we have 

^ = e 7i and e = --^ = (!-£) i 

whence it follows that 



tang. ^ = V^ /^HL^i^i! - 1 ; 
and the equation of condition for inclined floating is 



'>/. "" 



5 - '^ 6 e (1 - e)* 

Example 1) If the floating parallelopipedon is as high as wide, and if 
its specific gravity is t = |, we have 

tang. 6 = V2 VS . | — 1 = V3 — 3 = 1, whence (p = 45°. 

2) K the height h = 0,9 of the width h and the specific gravity is again 
^, we have 



tanrj. <p = V'3 . 0,81 -2 = V0,43 = 0.6557, whence = 33° 15'. 



756 GENERAL PRINCIPLES OF MECHANICS. [§372. 

§ 372. Specific Gravity. — The law of the buoyant effort or 
upward thrust of water can be made use of to determine the heavi- 
ness or specific gravity of bodies. According to § 364 the buoyant 
effort of the water is equal to the weight of liquid displaced ; hence, 
if we denote by V the volume of the body and by ji the heaviness 
of the liquid, we have the buoyant effort P = V y^. Now if ya is 
the heaviness of the material of the body, we have its weight 
G — F ys, whence the ratio of the heavinesses is 

y,_G_ 

I.E., the heaviness of the immersed tody is to the heaviness of the 
fluid as the absolute weight of the tody is to the huoyant effort or 
loss of weight during immersion. 

G F 

, Hence y^ = —- y-^ and yi = -— - yg, or if y denotes the heaviness 

of water, Cj the specific gravity of the fluid, and e^ that of the body,, 
we have, putting y^ = e^ y and y^ = £3 y, 

G , P 

^2 = -p £1 and ^i — -^ ^2. 

If we know the weight of a body and its loss of weight when 
immersed in a liquid, we can find from the heaviness or specific 
gravity of the liquid the heaviness and specific gravity of the ma- 
terial of which the body is composed, and, inversely, from the 
heaviness or specific gravity of the latter, the heaviness and specific 
gravity of the former. 

If the liquid in which we weigh solid bodies is water, we have 
£1 = 1 and yi = y .— 1000 kilograms = 62,425 pounds ; the former 
when we employ the cubic meter and the latter when we employ 
the cubic foot as unit of volume ; therefore in this case the heavi- 
ness of the body is 

G absolute weight i,. -,.-,,,-, , . « 

ya = -TT y = -1 — —5 ■ 7 , multiplied by the heaviness 01 water,. 

P loss 01 weight ^ "^ 

and its specific gravity is 

_ 6^ _ absolute weight 

P loss of weight 

In order to find the buoyant effort or loss of weight, we employ, 

as we do when determining the weight G, an ordinary balance. To 

the under side of one of its scale-pans is attached a small hook, from 

which the body is suspended by means of a hair, wire or fine thread 

before it is immersed in the water, which is contained in a vessel 

placed under the dish of the scale. A scale thus arranged for 



§372.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 757 

weighing under water is generally called a hydrostatic balance (Fr. 
balance hydrostatique ; Ger. hydrostatische Wage). 

If the body whose specific gravity is to be determined is less 
dense than water, we can combine it mechanically with some other 
heavy body, so that the compound mass will tend to sink in the 
water. If the heavy body loses in the water a weight P^ and the 
compound mass Pj, the loss of weight of the lighter body is 

P ^P^- P.- 

Now if G denotes the weight of the lighter body, we have its spe- 
cific gravity _ ^ _ ^ 



€2 = 



P P.- P.' 

If we know the specific gravity e of a mechanical combination 
of two bodies, and also the specific gravities £1 and £3 of the compo- 
nents, we can calculate from the weight G of the whole mass, by 
means of the well-known principle of Archimedes, the weights Gi 
and 6^2 of the components. 

We have G^ + G^— G, and also 

C C G 

volume — + volume —- — volume — ,or 

€iy £27- ey 









£2 


y • 


G, 


+ 


Gci 




G 


€1 




£2 




£ 



Combining the two equations, we obtain 

^.= ^(l_i):(i__l),or 

\£ £2 / V £1 £2 / 

^, = W1-1):(1-1). 

\£ £1 / \ £2 £1 / 

Example — 1) If apiece of limestone weighing 310 grams becomes 121,5 
grams lighter in water, the specific gravity of this body is 

_ ^1^ _ 

' - T2i;5 - ^'^^ 

2) In order to find the specific gravity of a piece of oak, a piece of lead 
wire, which lost 10,5 grams in weight when weighed in water, was wrapped 
around the piece of wood, which weighed 426,5 grams. The compound 
mass was 484,5 grains lighter in the water than in the air ; hence the spe- 
cific gravity of the wood is 

_ 426,5 _ 426,5 _ 

"~ 484,5 - 10,5 ~ 474 

3) An iron vessel completely filled with mercury weighed 500 pounds, 
and lost, when weighed in water, 40 pounds. If the specific gravity of the 
cast iron is = 7,2 and that of the mercury is = 13,6, the weight of the 
empty vessel is 



y58 GENERAL PRINCIPLES OF MECHANICS. [§373. 

«. = ^»» (m - w) ■■ (h - i) = '"" ^"'"' - '''''''^ -■ 

500 . 0,00647 3285 ,„ ^ 
(0,1388 - 0,0735) = —^-^^^g- = JSfi = ^^'^ ^'"""^'' 

and the weight of the mercury contained in it is 

500 . 0,0588 3940 
G., = 500 . (0,08 -- 0,1388) : (0,07353 - 0,1388) - — oo653"~ "" 653" 

= 450,2 pounds. 

Remark — 1) We can determine the specific gravity of fluids, loose 
granular masses, etc., by simply weighing them in the air ; for by enclosing 
them in vessels, we can obtain any desired volume of them. If the weight 
of an empty bottle is = G^ and when filled with water it is = G^, and if, 
when filled with some other liquid, its weight is (r^, the specific gravity 

of the latter liquid is 

G2 — G 
'"^ G^-G' 

In order, e.g., to obtain the specific gravity of rye (in bulk), we filled a 

bottle with grains of rye, and, after shaking it well, weighed it. After 

subtracting the weight of the bottle, that of the rye was found to be 

= 120,75 grams, and the weight of an equal quantity of water was 155,65 ; 

hence the specific gravity of the rye in bulk is 

120,75 ^ ^^^ 

— — := 776 

"155,65 '^' 

and a cubic foot of this grain weighs 

0,776 . 62,5 = 48,5 pounds. 

3) The problem, first solved by Archimedes, of determining from the 
specific gravity of a composition, and those of its components, the propor^ 
tion of each of the components, is of but very limited application to chem- 
ical combinations, alloys of metals, etc. ; for in such cases a contraction 
generally, and an expansion sometimes, takes place, so that the volume 
of the composition is no longer equal to ^the sum of the volumes of the 
components. 

§ 373. Hydrometers, Areometers. — We generally employ 
for tlie determination of the density of fluids areometers or hydrom- 
eters (Fr. areometres ; Ger. Araometer, Senkwagen). These instru- 
ments are hollow, symmetrical about an axis, have their centre of 
gravity very low down, and give, when we float them in any liquid, 
its specific gravity. They are made of glass, sheet brass, etc., and, 
according to the uses they are applied to, are called hydrometers, 
lactometers, salinometers, alcoholmeters, etc. There are two kinds 
of areometers, viz. : those with iveights (Fr. a volume constant ; Ger. 
Gewichtsaraometer), and the graduated areometers (Fr. a poids 



§373.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 759 



Fig. 




Fig. 684. 

A . 







constant; Ger. Scalenaraometer). The first are often used to de- 
termine the weight or specific gravity of sohd bodies. 

1) If F is the volume of the part of an areometer ABC, Fig. 
632, which is under water, when the latter floats vertically im- 
mersed to a point 0, G the weight of the whole apparatus, P that 
of the weights placed upon the dish A, when the apparatus floats 
in water, whose heaviness = y, and Pj their weight when the ap- 
paratus floats in another liquid whose density is yi, we will have 

Vy = P + G, 
Vy, = P, + G. 
Hence the ratio of the heavinesses or 
specific gravities of these liquids is 

yi _ Pi+ G 
y P + G' 

2) Let P be the weight, which must 
be placed upon the dish in order to im- 
merse the areometer ABC, Fig. 633, 
to a point 0, and let Pi be the weight, 
which must be placed upon the dish A 
with the body to be weighed in order 
to produce the same immersion, then 
we have simply 

G, = P- P,. 
But if we must increase Pj by P2 when 
the body to be weighed is placed in the 
lower dish G, which is under water, in 
order to preserve the same depth of im- 
mersion, the upward thrust is = P. 2 and 
the specific gravity of the body is 
_G^_ P -P , 
'-P,' P, • 
The hydrometer with the dish sus- 
pended below is employea for the de- 
1 3rmination of the specific gravity of 
solid bodies, such as minerals, etc., and 
is called Nicholson^s hydrometer. 

3) If we put the weight of an areom- 
eter B G with a graduated scale A B, 
Fig. 634, = G, and the immersed vol- 
ume, when it floats on water, = F, we have 6^ = F y. If the 
areometer rises a distance OX— x, when immersed in another 



Fig. 633. 




760 GENERAL PRINCIPLES OF MECHANICS. [§37?. 

liquid, we have, when the cross-section of the shaft is denoted by 
F^ the volume immersed 

= F — F X, and therefore G — ( F — F x) y^. 
Dividing the two formulas by each other, we obtain the heaviness 
of the liquid 

F i, ^ \ y 



'\l-yx) = - 



Fx ' ' \ VI 1-iiQf 

F 
in which fi denotes the constant quotient ^. 

If the liquid in which the areometer floats is lighter than water, 
it will sink in it a distance x, and we will have 

6^ = ( F + F x) y, and therefore 

r 

Ti — ^ — • 

' 1 -^ lix 

F . 

In order to find the coefficient [i = ^, we increase its weight 

by an amount P, e.g. by pouring mercury in the areometer at the 
upper end, so that it passes to the bottom of it, rendering the ap- 
paratus so much heavier that, when floating in water, a consid- 
erable portion of the length of the stem, to which the scale is 
applied, is immersed. Putting P = F ly,l denoting the immer* 
sion produced by P, we obtain 

_F^_ ^_ _ P_ 
^ ~ V ~ Vly ~ GT 

Example — 1) If an areometer, weighing 65 grams, must have 13,5 
grams removed from the dish in order to float at the same depth in alcohol 
as it had done in water, the specific gravity of alcohol is 

= — ~^ ' = 1 - 0,208 = 0,793. 

3) The normal weight of a Nicholson hydrometer is 100 grams ; that 
is, we must place 100 grams upon the dish in order to sink the instrument 
to 0, but we must take away 66,5 grams after having laid a piece of brass 
which we wish to weigh upon the upper dish, and 7,85 grams had to be 
added when the brass was removed to the lower dish. The absolute weight 
of the brass is then 66,5 grams and its specific gravity is 

'-'-''==847 

7,85 ' 

3) A graduated areometer, weighing 75 grams, rises, after 31 grams of 
the substance, with which it was filled, has been removed, a distance 2 = 6 
inches = 73 lines ; the coefficient // is therefore 

31 

= ^^1^ ^- 0,00574. 
75 . 73 ' 



374.] EQUILIBRIUM OF WATER WITH OTHER BODIES. 



761 



It was then refilled until its weight became again 75 grams, when it was 
placed in a solution of salt ; it fhen rose a distance of 29 lines ; the specific 
gravity of the liquid is therefore 

= 1 : (1 - 0,00574 . 39) = 1 : 0,833 = 1,2. 

Remark. — A more extended treatment of this subject belongs to the 
province of chemistry, physics and technology. 

§ 374. Liquids of DiflFerent Densities. — If several liquids 
of different densities exist in a vessel at the same time without 
exerting any chemical action upon one another, they will place 
themselves, in consequence of the ease with which their particles 
are displaced, above each other in the order of their specific gravi- 
ties, viz : the most dense at the bottom, the less 
dense above it and the least dense on top. When 
in equilibrium the limiting surfaces are hori- 
zontal ; for so long as the limiting surface E F 
between the masses Jf and N, Fig. 635, is inclined 
so lorlg will there be columns of liquid, such as 
G K, Gx K^, etc., of dilTerent weights above the 
horizontal layer H R ; hence the pressure upon 
this layer cannot be the same everywhere and 
consequently equilibrium cannot exist. 

The liquids arrange themselves also in the communicating tubes 
A B and C D, Fig. 636, according to their specific gravities above 
one another, but their surfaces A and D G do not lie in one 
and the same horizontal plane. 



Fig. 635. 




Fig. 636 




Fig. 637. 
Gx«^D E 




If i^is the area of the cross-section IT R of a piston, Fig. 637, 

in one leg ^ ^ of two communicating tubes and h the head of 

water or the height E H oi the surface of the water in the second 

tube C D above H R, we have the pressure upon the surface of the 

piston 

P = Fhy. 



762 GENERAL PRINCIPLES OF MECHANICS. [§375,376. 

If we replace the force, exerted by the piston, "by a column of 
liquid H A R, Fig. 636, whose height is h^ and whose heaviness 
is yi, we have 

equating the two expressions we obtain 

Ih Ti = ^ % 
or the proportion 

]h ^ y_ 
h yi 
TJierefore tlie heads or the heights of the columns of liquid, 
measured from the common lolane of contact of ttvo different liquids^ 
which are in equilihrium m two communicati7ig tithes, are to each 
other inversely as the heavinesses or specific gravities of these liquids. 
Since mercury is about 13,6 times as heavy as water, a column 
of mercury in communicating tubes will hold in equilibrium a 
column of water 13,6 times as long. 



CHAPTER III. 

OF THE MOLECULAR ACTION OF WATER. 

§ 375. Molecular Forces. — Although the cohesion of water 
is very slight, it is not null. The molecules (Fr. molecules ; Ger. 
Theile or Molekiile) not only cohere together, but also adhere to 
other bodies, e.g., to the sides of a vessel, so that a certain force is 
necessary to destroy this union, which we call the adhesion (Fr. ad- 
herence ; Ger. Adhasion) of the water, A drop of water, which 
hangs from a solid body, demonstrates the existence of the cohe- 
sion and of the adhesion of the water. Without cohesion the 
water could not form a drop and without adhesion it could not 
remain hanging from the solid body ; gravity is here overcome not 
only by the cohesion, but also by the adhesion. The actions, arising 
from the combination of the forces of cohesion and adhesion, are 
called, to distinguish them from the actions of inertia, of gravity, 
etc., the molecidar actions. Capillarity or the raising or depressing 
of the surface of water or mercury in narrow tubes or between plates, 
placed close together, is an important instance of molecular action. 

§ 376. Adhesion Plates. — The cohesion and adhesion of 
water have been determined by means of adhesion plates. To 



§ 377.] THE MOLECULAR ACTION OF WATER 763 

accomplish this object, such a plate is suspended (instead of the 
scale pan) af one end of the beam of a balance, which is brought 
into equilibrium by means of weights ; the vessel containing the 
liquid to be examined is then caused to approach gradually, until 
the surface of the liquid comes in contact with the plate. Weights 
are now gradually placed upon the dish at the other end of the 
beam, until the plate is torn away from the surface of the water. 
The results of such experiments depend particularly upon the fact 
whether the plate is moistened by the water or not. In the first 
case after the contact a thin sheet of water remains hanging to the 
plate ; hence in tearing the latter from the water, we overcome not 
the adhesion, but the cohesion of the water. Hence the force 
necessary to tear different plates from the surface of the water 
does not depend upon the nature of the material, of which 
the plates are composed. Other liquids, on the contrary, require 
different forces to be applied to the adhesion plates. Dtt Buat 
found that the adhesion between water and tin plate was from 65 
to 70 grains per square inch (old Prussian measure). This gives a 
force of about 5 kilograms for a square meter, or 1,024 pounds per 
square foot. Achard found values differing but little from the 
above for lead, iron, copper, brass, tin and zinc. Gay Lussac ob- 
tained the same results v/ith a glass disc, and Hiitli with different 
kinds of wooden plates. 

If, on the contrary, the surface of the disc is not moistened by 
the surface of the water, the results obtained are totally different ; 
for in this case it is not the cohesion, but the adhesion of the water 
which is overcome. It appears that the duration of contact has a 
great influence upon the force necessary to tear the disc loose, e.g.. 
Gay Lussac found that, with a glass plate 120 millimeters in diam- 
eter, a force varying from 150 to 300 grams, according as the dura- 
tion of contact was long or short, was necessary to tear it loose from 
a surface of mercury. 

Remark. — In Frankenbeim's "Lehre der Cohasion'' the phenomena of 
cohesion, as, e.g., those presented when moistened plates are torn from the 
surface of water, are called " Byna'phy^'^ and, on the contrary, the phenomena 
of adhesion, as, e.g., those occurring during the separation of unmoistened 
plates from the surface of a liquid, " Prosaphy^ 

§ 377. Adhesion to the Sides of a Vessel. — If a drop 
of water spreads itself out upon the surface of another body and 
moistens it, the adhesion is in this case predominant; but if, on 



764 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 377. 



the contrary, the drop retains its spherical form upon the surface 
of a solid or fluid body, the cohesion is the strongest. The com- 
bined action of these two forces upon the surface of a liquid near 
the walls of the vessel is particularly remarkable ; .the water rises 
up and forms a concave surface when the cohesion is less powerful 
than the adhesion, and the wall becomes moistened in consequence ; 
the surface of the water, on the contrary, is curved downwards in 
the neighborhood of the walls of the vessel and forms a convex 
surface when the side of the vessel is not moistened or when the 
cohesion is predommant. 

These phenomena can be easily explained as follows. 

A molecule E in the surface H R oi the water (Fig. 638) is 
drawn downwards in all directions by the surrounding water, and 
the resultant of all these attractions is a single force A acting ver- 
tically downwards ; on the contrary, a molecule E at the vertical 
wall B E, Fig. 639, of the vessel is acted upon by the wall with a 

Fig. 638. Fig. 639. 





Fig. 640. 



horizontal force P and by the water filling the quadrant B E 
with a force A, whose direction is inclined downwards to the hori- 
zon ; the direction of the resultant R of these two forces is at 
right angles to the surface of the water (see § 354). According as 
the attractive force of the wall of the vessel is greater or less than 
the horizontal component Ai of the mean force of cohesion A of 

the water, the resultant R will assume a di- 
rection either from without inward or from 
within outward. In the first case (Fig. 639) 
the surface of the water at E rises along 
the wall, and in the second case it descends 
along the wall B E, as is represented in Fig. 
640. 
These relations change, when the water reaches to the brim of 
the vessel ; for the direction of the attractive force of the wall of 
the vessel is then different. If, E.G., the surface of the water E 0, 
Fig. 641, which in the beginning reached to the brim C of the vessel 
B C 0, is caused to rise gradually by adding water, the inclination 
of the force of adhesion to the horizon will gradually increase, and 




Vy^tyV/W//M//,//////,/^/,,,//M,/,y/,/„„„„. 



378.] 



THE MOLECULAR ACTION OF WATER. 



765 



its horizontal component will, in consequence, gradually decrease, 
until it becomes less than the horizontal component Ai of the force 
of cohesion A. Consequently the form of the surface of the water 



Fia. 641. 




at U changes continually, until its con- 
cavity becomes a convexity and the de- 
pression below the brim of the vessel be- 
comes an elevation, which must attain a 
certain height before the water will flow 
over the side of the vessel. 



§ 378. Tension of the Surface of the "Water. — Since each 
molecule in the surface H R, Fig. 638, of a liquid is attracted down- 
wards by the mass of liquid below it with a force A, we can assume 
that a condensation and a coherence among the molecules of the 
liquid upon the surface will be the result and that a certain force 
will therefore be necessary to overcome this coherence or to tear 
the surface of the liquid. This coherence of the surface of a liquid 
shows itself not only whenever a foreign body is dipped into it, 

but also whenever the surface 
Fig. 642. 




Fig. 643. 




of the liquid becomes curved, 
as, E.G., m the neighborhood 
of the wall of the vessel. If 
we assume with Young that 
the tension or cohesion of the 
surface of a liquid is the same 
in all parts of it, we can de- 
duce, as GeJieimer Oberhau- 
ratli Hagen has proved, from 
that hypothesis all the laws 
of capillary attraction which 
coincide best with the results 
of experiment. 

In the neighborhood of a 
plane wall D G, Figs. 642 and 
643, the surface of the liquid 
forms a cylindrical surface 
D A H, which is convex either 
upwards or downwards. If P 
is the normal force upon an 
element A E B = g of this 
surface, 8 the tension of this 



766 



GENERAL PRINCIPLES OF MECHANICS. 



[§379. 



element and r its radius of curyature A — C B, wq have, in 

consequence of the similarity of the triangles EPS and ABO, 

P A B a 
Fig. 644. 




C>^^.--^.XX>vXNNS\\\\NXXXN\NN\\\X.\^.\XX\N.s\^^^^^ 



Fig. 645. 




S CA r' 
and, therefore, the normal or 
bending force is 

r 
!N"ow if the element A E B 
of the surface is at the vertical 
distance R = y above or 
below the surface of the water 
which is free from the influ- 
ence of the wall D G, and if y 
denotes the heaviness of the 
liquid, we have, according to 
(§ 356) the well-known law of 
hydrostatics, the pressure of 
the water upon the element 

ATB = o 

P = oyy; 
we can therefore put 

a y y = - S and 



y = 



_s_ 

r y 



Hence the depression or elevation of an element of the surface 
of a liquid in reference to the free or unaffected part of this surface 
is inversely proportional to the radius of curvature. 

§ 379. In the vicinity of a curved' ivall, e.g., of a vertical cylin- 
drical surface, the surface of the water forms a surface of double 
curvature and the column of water below the rectangular element 
F G H K, Fig. 646, of the surface is solicited by two forces Pi and 
Psj one of which is the resultant of the tensions S^, S^ in the nor- 
mal plane ABE, parallel to the side F G — H K; the other is 
the resultant of the tensions S^^, 8^ in the normal plane ODE, 
parallel to the side G H = F K. The former plane corresponds 
to the greater and the latter to the least radius of curvature ; put- 
ting the two radii = r^ and rs and the length of the sides F G = o^ 
and (} H — a^_ and denoting the tension for a width = unity by S, 
we have the tensions acting in the two planes 



380.] 



THE MOLECULAR ACTION OF WATER. 



767 



Si = (72 S and S^ = o^ S 
and the normal forces resulting from them 

<7^ S Cj (Tg 






P. = (y,S 



n 



S Oi G^ 



and 



, and their resultant is 



P^P, + P,= 8a,aA— + — ). 



Fig. 646. 




If here also y denote the 
height of the element FGHJ^ 
of the surface (which may be 
regarded as a rectangle, whose 
area is (Tj g^) aboye the low- 
est or general surface of the 
water, we have the force, with 
which this element is drawn 
normally upwards or down- 
wards by the water above or 
below it, 

P = y G.G.y, 
equating the two values for 
P, we obtain 



y G^ G^y = S Gi G<i I — ■ -\ ), whence 



y 






When the wall is cylindrical the elevation (depression) of the 
surface of the water above (below) the general water level is at 
every point proportional to the sum of the reciprocals of the maxi- 
mum and minimum radii of curvature. This formula contains 
also that of the foregoing paragraph ; for if the normal section 
C B D is Si right line, we have 

whence 



00 



— = and 

S 1 
y = - . — . 



(§ 380.) Curve of the Surface of Water— The curve 
formed by the vertical cross-section of the surface of the water 



768 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 380. 



Fig. 647. 



near a plane wall, can be found, according to Hagen, in the follow- 
ing manner. Let A R, Fig. 647, be the surface of the water 

attracted by the vertical wall B E, 
H R the general level of the water, 
and let the point of intersection H of 
the two surfaces be the origin of co- 
ordinates. Let us put the co-ordinates 
of a point of the surface A R,H M 
— X and M = y, the arc A — s^ 
the tangential angle T M — a, and 
the elements Q, Q P and P re- 
spectively = d Xy d y and d s. 

cr 

Since y = — , and, according to 

Article 33 of the Introduction to the 
Calculus, 

r = — ^f— and d y = — d s sin. a, we have 
da ^ 




y = - 



8 d a S sin. a . d a 



y ds 

8 



y dy 



or 



y d y ^= ~ sin. a . d a, 
by integrating which we obtain 

i y = — / sm. a . d a = 



Con. 



8 
7 



cos, a. 



Since for the point R, a and y are both = 0, we have 



a a 

= Con. COS. 0, whence Con. = — and 

y r 



f 

hence 



r 



COS. a) 



4.8 {1 



y 



r 



r 



sm. ^ a. 



- COS. a) 4: 8 , . , v„ 



For a = 90°, we have sin. i a = sin. 45° = V^ ; hence tlie 
maximum elevation of the water immediately against the wall is 



8 



h = 2 y-- ' Vi 

7 

— = ^ ^^ and 
7 



A 



8 



, or inversely 



1) y = h V^ . sin. ^ a. 



§ 380.] THE MOLECULAR ACTION OF WATER. 769 

Differentiating this expression, we obtain 

d y = Ih V'^ COS. ^ a . d a — h V}f cos. \ a , d a, 
and since d y = — d x . tang, a, it follows that 

, ,/Y COS. A a 7 7 ,/T COS. ^ a COS. a _ 

dx = — h Vi . = — . d a = — k V-h . —. . a a 

tang, a " sm. a 

J ,/7 COS. .1 a \(cos.^ aV — (sin. i aVl _ 

= — hVi . ^-^-- — f — h ^—^^ d a 

^ 2 sm. -^ a . COS. -| a 

^ .nr 1 — 2 {sin. ^ a-Y -, 
^ 2 sm. i a 

= — h VJ . I . '~ , sin. I a) d a, 

^ \sm. \a ^ I 

But now 

/ sin. I a , d a =: — 2 cos. | a and 

/—. — z — = 21 tana. \ a 
sin. ^a ^ ^ 

(see Introduction to the Calculus, Art. 29) ; 

hence we have 

z = — h V^ (l tang. | a + 2 cos. j a) + Con. 

Now since for x = 0, a° = 90% tang. ^ a = tang. 22^° = V^ — 1 
and cos. J a = ^2, it follows that 

Con. = hVl[l{V2 - 1) + 2 VI], and 

= h[l - V2. COS. ia - Vll{V2 + 1) tang, | a]. 
For a = we have 

COS. I a = 1 and I tang, \ a = — 00 , 
and therefore 

a; = + CO ; 
ff E is consequently the asymptote, which the section A R oi 
the surface of the water continually approaches. 

Remabk. — If we invert the formula (1) and put 

sin. ^ a = ^ Vf 

we can calculate for every value of y, first a and then by means of (2) the 
corresponding value of x. 
49 



770 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 381. 



The measurements made by Hagen to test this theory, show that it 
agrees very well with the results of experiment. They were tried with a 
dead polished brass plate upon spring water, and gave the following results. 



y measured in lines 
\x calculated 



1,37 
0,00 
0,00 



0,70 
0,31 
0,33 



0,49 
0,63 
0,64 



0,34 
0,94 
0.96 



0,24 
1,26 

1,28 



0,18 
1,57 
1,56 



0,12 

1,88 
1,95 



0,07 
2,50 

2,47 



0,04 
3,13 
3,01 



0,016 

3,74 

3,90 



Fig. 648, 



D 



H= 



IS 



G 



D 



These values are given in Paris lints. From A = 1,37 lines we calcu- 

late — = 0,94 and the minimum radius of curvature r = 0,68 lines. Plates 

7 
of boxwood, slate, and glass gave the same results. 

§ 381. Parallel Plates. — The water between two plates 
D E, D E, Fig. 648, ivhicli are placed near each other, rises not 

only on the outside, but also between them 
and the cross-section of its surface is nearly a 
semi-ellipse. One semi-axis of the elliptical 
cross-section is the half width C A = a, the 
oth«r semi-axis C B = b is equal to the dijffer- 
ence A F — B G = h.^ — h^ of the maximum 
and minimum elevations of the elliptical sur- 
face A B A above the general water level. 
According to the " Ingenieur," page 171, the 
radius of curvature of the ellipse at A is 
b' ^ {h, - h,y 
a 






mm 



F^R 



mfrrrrriffffffi 



^,.,„,,„„„„,i 



^M 



r, — — 



T 



a 

a' 



, and that at B is 



{h, -h,)' 

hence we have, according to § 378, the elevation of the surface of 
the water at A 



h = — = 



a S 



and, on the contrary, that at B 

r^y a' y 



h 



Subtracting the latter equation from the former, we obtain 

8 / a hi — hA 



h 



or 



8 I a 1 \ 

r \(^. - h,) ^ ^ ?> 



a' 



1 = ^ 



§381.] THE MOLECULAR ACTION OF WATER. 771 

whence 

1) n. 



— hi = a y 



s + a' y 



2) h = ' V- (^ + 4. 
« y \ y / 



3) /, = 1^ V^ '^ 



a ' y ' jS + a" y^ 
and, finally, the ratio 

_ 7^2 — hi _ «' y _ 2 . ^ 
hi S y 

If a is very small, we can put 

h.2 = hi = - . — 
a y 

tJie elevation of the surface of the ivater is then inversely proportional 

to the distance of the plates from each other. 

We have, however, more accurately, 



, I S L , a'y\ 1 S ^ 
a y \ ^ 8 J ay ^ 

a y \ "^ 8 J 



-la. 



y \ o / ay 

1 8 L , a'yX 1 ^ 

a ' y 

By inversion we obtain 

8 ^ a' 

- = ahi + -. 

These formulas agree very well with the results of observation, 

especially when -j- does not reach ^. 
fii 

Hagen found, from his experiments with two parallel plane 

plates in spring water, as a mean 

hi = 1,55, h.2 = 2,09, and h = 1,38 Paris lines, 

and by calculation 

a 

— = 1,04, A2 = 2,12, and h — 1,44 Paris lines. 

More recent experiments (see Poggendorff's Annalen, Vol. 77) 
gave for 

a = 0,360 ; 0,5875 ; 0,7575 lines, 
hi= 2,562; 1,429; 1,068 lines, and 

— = 0,949 ; 0,907 ; 0,917 lines, 



772 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 382. 



I.E. as a mean value 

~ = 0,9243 and S = 0,01059 grams. 

(Compare the foregoing paragraph.) 

§ 382. Capillary Tubes. — We can easily calculate the height 
to which the surface of water will rise in narrow vertical tubes, 
called capillary tubes (Fr. tubes capillaires ; Ger. Haarrohrchen), by 
starting from the formula 

S /I 1\ 

2/ = — — + — ) 

^ y \ri rj 

of § 379 as a basis and assuming that the sur- 
face (the meniscus) forms a semi-spheroid 
ABA, Fig. 649, whose circular base A A coin- 
cides with the cross-section of the tube. If we 
retain the notations of the foregoing paragraph, 
I.E. if we put the radius C A of the tube = a 
and minimum and maximum heights B G 
and A F of the water in the tube above the 
general level of the water H B, = hi and h^, 
we must substitute in 



Fig. 649. 
D D 




7. ^ /I 1\ 

flu = — ( \- —J, Ti = a and r^ 

^j = ( — + — l n = ^2 

y Vr, rJ 



a 



-, and in 



h = — - + 



h — hi 

a 



h, = 



y \a (^2 — hi) 

2 S. (h, - hi) 



7> 



thus we obtain 



and 



y a 

Subtracting the last equation from the one preceding it, we 
obtain 

a 2 (As — hiY 

■y \a ~^ '^ 
or 



h 



7 'S /I 
y \a 



{h, - hiY a' 

y \a {h, - hi) "^ (h, - hiY a' F 

If a is smalL we can put 



and also 



§ 382.] 



THE MOLECULAR ACTION OF WATER. 



773 



2 1 

-X (^2 ~ hiY - - (^2 - h,Y = a, 

whence it follows that 

hi — hi = ay 
assuming h^ — h^ = a + 6 and putting {h^ — h^)'^ = a^ + 2 a 6, 
and also (h^ — h^f = a^ -[■ ^ a^ cJ, we obtain 

[\ + -^) (^' + 3 «M) - - («' + 2 «^ d) = «, ^ 

\ h Cb I Ob 



or 



whence it follows that 

^ = — T. ¥ r-FT? or approximatively, (5 = — 4—^. 

3 y«' + 4/S^ ^^ ^ 4:S 

Hence we have 
whence 



/2'2 ill — ^ A Clf 



y a 



2iS'l/ ya\ 2 S a . 
. —^ [a — j-^ ) = - . and 

«M 4 ^/ 



S /I 
h, = —I + 

7 I ^ 



a 



a ' y 2 
S 



( 7 ci'\ 
~ y U "^ ^ l^ "^ 2 /SVJ - ^ • y "^ 2' 



a'^ a'v'^^s)} 



y \-a a \ 4 o/j a y 

The mean elevation in capillary tubes is inversely proportional 
to the width of the tube. 

We have also for the determination of S the formula 

8 , ^ a" 

— = h ahi + ~. 

y 4 

Observations made by Hagen with capillary tubes in spring 
water gave the following results : 



Width of tube a, lines 


0,295 


0,336 


0,413 


0,546 


0,647 


0,751 


0,765 


Elevation A^, " 


10,08 


8,50 


6,87 


5,17 


4,28 


3,72 


3,59 


Measure of) 8 
tension > y ^ 


1,508 


1,455 


1,458 


1,478 


1,473 


1,512 


1,494 



774 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 383. 



According to these experiments the mean yalues are 

— = 1,482 and .S^ = 0,0170 grams. 

The variations in these values are due to the fact that the ten- 
sion S of the surface of the water diminishes with the time, and is 
much smaller in water that has been boiled, than in fresh. We 
can now assume that the tension of the water in every strip 1 line 
wide is S = 0,0106 to 0,0170 grams. 

§ 383. The foregoing theory is also applicable, 2vhe7i the wall is 
not moistened by the liquid ; here, however, it is not an elevation 
but a sinking of the surface which takes place, and the latter is 
concave instead of convex. The vertical force P, which is due to 
the difference of level B G and acts from below upwards, is balanced 
by the tensions S and S of the surface ABA, Fig. 650, of the 
liquid in the tube. The force of adhesion of the solid body does 
not, according to the foregoing theory, come into play in this case. 

Fig. 650. 



D D 





If we make the force, with which the wall of the tube attracts 
to itself the column of fluid B G, Fig. 651, proportional to the 
circumference of the tube, if, e.g., for a cylindrical tube we put this 
force P — M 2 TT a, in which fi denotes a coefficient, we have 

7T d^ h = 2 iJ- TT a, 
and, therefore, the mean elevation of the water in the tube is 

k = ^'i. 

a 
For two parallel plates, on the contrary, we have P = 2 fi I and 
P = 2 a h I y, I denoting the undetermined length of the column 
of water, and, therefore, 



h = 



a' 



I.E., half as great as in a tube3 when the distance 2 a of the plates 



§383.] THE MOLECULAR ACTION OF WATER. 775 

from each other is equal to the diaraeter of the tube. This agrees 
also with the results of the last paragraph. 

According to HagerCs experiments the strength or tension of 
the surface of liquid does not depend upon its degree of fluidity, 
but it increases in intensity, the more the liquid adheres to other 
bodies. According to others, particularly Brunner and Franken- 
Jieim (see Poggendorf 's Annalen, Vols. 70 and 72), the height h, to 
which water rises in capillary tubes, increases and S consequently 
diminishes, when the temperature of the liquid is augmented. For 
alcohol ;iS' is about one-half and for mercury about eight times the 
strength of the surface of water. 

Remark — 1) Hagen found by measuring and weighing drops of liquid, 
which tore themselves loose from the base of small cylinders, about the 
same values as he did by Ms observations upon cajDillary plates. In like 
manner the experiments with adhesion plates have furnished results, v^hich 
coincide very well with the former, when we assume that the force neces- 
sary to tear the plate loose is balanced by the weight of the cylinder of 
liquid raised and by the tension uj)on the surface of this cylinder. 

2) The number of treatises upon capillary attraction is so great that we 
cannot cite them all here. The greatest mathematicians, such as La Place, 
Poisson, Gauss, etc., have given their attention to it. A complete account 
of the older literature is to be found in Frankenheim's " Lehre von der Co- 
hasion." The treatise which was specially used in preparing this chapter is 
the following: "Ueber die Oberflache der Flussigkeiten," by Hagen, a 
memoir read in the Royal Academy of Science in Berlin, in 1845. A new 
physical theory of capillary attraction, by J. Mille, is contained in Vol. 45 
ot Poggendorff's Aunalen (1838). Here also belong Boutiguy's Studies 
of Bodies in a Spheroidal Condition. 



776 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 384 



CHAPTER IV. 



Fm. 652. 



OF THE EQUILIBRIUM AND PRESSURE OF THE AIR. 

§ 384. Tension of Gases. — The atmospheric air, which sur- 
rounds us, as well as all other gases (Fr. gaz ; Ger. gase) possess, in 
consequence of the repulsion between their molecules, a tendency 
to expand into a greater space. We can therefore obtain a limited 
quantity of air only by enclosing it in a perfectly tight vessel. The 
force with which the gases seek to expand is called their tension 
(Fr. tension ; Ger. Spannkraft, Elasticitat or Expansivkraft). It 
shows itself by the pressure exerted by the gas upon the walls of 
the vessel enclosing it, and differs from the elasticity of solids or 
liquids in this : it is in action, no matter what the density of the 
gas may be, while the expansive force of solids and 
liquids is null, when they are extended to a certain de- 
gree. The pressure or tension of the air and other 
gases is measured by haromsters, manometers and valves. 
The harometer (Fr. barometre ; Ger. Barometer) is em- 
ployed principally to measure the pressure of the atmo- 
sphere. The most common kind is the so-called cistern 
harometer, Fig. 652 ; it consists of a glass tube, closed 
at one end A and open at the other B, which, after be- 
ing filled with mercury, is turned over and placed with 
its open end under the mercury contained in the vessel 
C D. After the instrument has been inverted, there 
remains in the tube a column B S of mercury, which 
(see § 374) is balanced by the pressure of the air upon 
the surface II R. Since the space A S above the col- 
umn of mercury is free from air, the column has no 
pressure upon it from above, and the height of this 
column, or rather that of the mercury in the same, 
above the level H B of the mercury in the vessel can 
be employed as a measure of the pressure of the air. 
In order to measure easily and correctly this height, 
an accurately graduated scale is added, which can be 
moved along the tube and which is sometimes provided with a 
movable pointer S. 



■ I 




§385.] EQUILIBRIUM AND PRESSURE OF THE AIR. 777 

Remakk. — It is the province of physics to give more detailed descrip- 
tions of different barometers, to explain their use, etc. (See Miiller's Lehr- 
buch der Physik und Meteorologie, Vol. I.) 

§ 385. Pressure of the Atmosphere. — By means of the 
barometer it has been found that in places situated near the level 
of the sea, when the atmosphere is in its average condition, the 
pressure of the air is balanced by a column of mercury at a tem- 
perature of 32"' Fahr., 76 centimetres long or about 28 Paris inches 
= 29 Prussian inches = 29,92 English inches. Since the specific 
gravity of mercury at 32° temperature is 13,6, it follows that the 
pressure of the air is equal to the weight of a column of water 
0,76 . 13,6 = 10,336 metres = 31,73 Paris feet == 32,84 Prussian 
feet = 33,91 English feet. We often measure the tension of the 
air by the pressure upon the unit of surface. Since a cubic centi- 
metre of mercury weighs 0,0136 kilograms, the atmospheric pres- 
sure or the weight of a column of mercury 76 centimetres high, the 
base of which is 1 square centimetre, is 

p = 0,0136 . 76 = 1,0336 kilograms. 

But a square inch is 6,451 square centimetres, and therefore the 
mean pressure of the air is also measured by 1,0336 . 6,451 = 6,678 
kilograms = 14,701 pounds upon a square inch = 2116,9 pounds 
upon a square foot. Assuming the exact height of the barometer 
to be 28 Paris inches = 29 Prussian inches, we obtain for the 
pressure of the air upon one square inch 14,103 Prussian pounds 
and upon the square foot 2030 Prussian pounds. 

The standard usually adopted, where the English system of 
measure is used, is 14,7 pounds upon the square inch, which cor- 
responds to a column of mercury about 30 (exactly 29,922) inches 
and to a column of water about 34 (exactly 33,9) feet high. It is 
very common in mechanics to take the pressure of the atmosphere 
as the unit and to refer other tensions to it; they are then given in 
pressures of the atmosphere, or simply in atmospheres. Thus a 
column of mercury 30 . n inches high, or a weight of 14,7 . n Eng- 
lish pounds, corresponds to the pressure of n atmospheres, and, in- 

Ji 
versely, a column of mercury h inches high to a tension ^- = 

0,03333 h atmospheres and the tension -r— = 0,06803 p atmo- 
spheres to a pressure of ^ pounds upon a square inch. Besides the 
equation ^f.-^^^ = 77^ gives the formulas for reduction 

ZuyjZZ 14, I 

h — 2,0355 p inches and p — 0,4913 h pounds. 



778 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 886. 



For a tension of h inches — j^ pounds the pressure upon a sur- 
face of F square inches is 

P = Fj) = 0,4913 Fh pounds 
= Fhy = 2,0355 F j) inches. 
Example — 1) If the level of the water is 350 feet above the piston of a 
water-pressure engine, the pressure upon the piston is 

= -^ = 7,4 atmospheres. 

2) If the air in a blowing-cylinder has a tension of 1,3 atmospheres, the 
pressure upon every square inch of the same is 

= 1,3 . 14,7 = 17,64 pounds, 
and upon the piston, whose diameter is 50 inches, 

-J-- . 17,64 = 34636 pounds. 

50^ 



7T 



14,7 = 38863 



Fig. 653. 



Since the atmosphere exerts an opposite pressure 

lbs., the force of the piston is 

P = 34636 - 38863 = 5773 pounds. 

§ 386. Manometer. — In order to determine the tension of 
gases or vapors which are enclosed in vessels, we employ instru- 
ments, which resemble barometers and are called ma- 
nometers (Fr. manometres ; Ger. Manometer). These 
instruments are fdled with mercury or water and are 
either open or closed ; in the latter case the upper part 
may be free from air or filled with it. The manome- 
ter tvitJi a vacuum, above the column of mercury, as is 
represented in Fig. 653, is like the common barometer. 
In order to be able to measure with it the tension of 
the air in a gasholder, a tube C F is added to it, one 
end of which C opens into the gasholder and the other 
end F enters above the level of the mercury H R into 
the case H D R oi the instrument. The space H F R 
above the mercury is thus put in communication with 
the gasholder ; the air existing in this space assumes 
the tension of the air or gas in the gasholder and 
presses a column of mercury B S into the tube, which 
balances the tension of the air that is to be meas- 
ured. 

The syplion manometer ABC, Fig. 654, which is 
open at the end A, gives the excess of the tension of the 
gas in a vessel above the pressure of the atmosphere ; 
for that tension is balanced by the combination of the pressure of 
the atmosphere upon ^S' and of the column of mercury R S. If 5 




§ 386.] 



EQUILIBRIUM AND PRESSURE OF THE AIR. 



779 



is the height of the barometer and h that of the manometer, or the 
distance R 8 between the surfaces ^and S of the quicksilver in 
the two legs of the manometer, the pressure of the air which is in 
communication with the short leg will be expressed by the height 
of the column of mercury 

or by the pressure upon a square inch 

p = 0,4913 {b + 7i) pounds, 
or, if b is the mean height of the barometer, 

p ^ 14,7 + 0,4913 h pounds. 

The cistern manometer A B C D, Fig. 655, is more common 

than the syphon manometer. Since in the former the air acts 

upon the column of liquid through the medium of a large mass 

of. mercury or water, the vibrations of the air are not so quickly 

Fig. 654 Fig. 655. Fig. 656. 






communicated to the column of liquid, and consequently the meas- 
urement of the column, which is less agitated, can be made more 
easily and more accurately. In order to facilitate the reading of 
the instrument, a float, which communicates by means of a string, 
passing over a pulley, with a pointer, which is movable along the 
scale, is often placed on top of the mercury in the tube. 

Manometers can also be used for the purpose of measuring the 
pressure of water and other liquids ; in this case they are called 
piezometers (Fr. piezometres ; Ger. Piezometer). 

By the aid of a valve D E, Fig. 656, the tension of the gas or 
steam, contained in a vessel M N, can be determined, although not 
with the same accuracy, by placing the sliding weight G in such a po- 
sition that it balances the pressure of the steam, li C S = 5 is the 
distance of the centre of gravity of the lever from the axis of rota- 
tion C, C A — a the arm of the lever of the sliding weight and Q 
the combined weight of the valve and lever, we have the statical 
moment, with which the valve is pressed downwards by the weights, 



780 GENERAL PRINCIPLES OF MECHANICS. [§ 387, 

= G a — Q s; 

now if the pressure of the gas or steam upwards = F, the pressure 
of the atmosphere downwards = Pi and the arm of the lever C B 
of the valve = i, we have the statical moment with which the 
valve tends to open 

= (p - p.) b, 

equating the two moments, we obtain 

P d — Pxh = G a + Q s, and consequently, 

Ga ■¥ Q s 

If r denote the radius of the valve D E, p the interior and p^ 
the exterior tension, measured by the pressure upon a square inch, 
we have P ^ -n r" p and P^ = tt r"" px^ whence 

G a + Qs 

P =P\ ■\ TT — • 

Example — 1) If the height of the mercury in an open manometer is 
3,5 inches and that of the barometer 30 inches, the corresponding tension is 
A = & + Aj z= 30 + 3,5 = 33,5 inches, or 
p = 0,4913 . ?i = 0,4913 . 33,5 = 16,46 pounds. 

2) If the height of a water manometer is 21 inches and that of the 

barometer is 29 inches, the corresponding tension is 

21 
h = 29 + -T^ = 30,54 inches = 15,0 pounds. 

3) If the statical moment of a safety valve, when not loaded, is 10 inch- 
pounds, if the arm of the lever of the valve, measured from the valve to 
the axis of rotation, is 5 = 4 inches and its radius is r = 1,5 inches, the 
difference of the pressures upon the valve is 

150 + 10 160 
P-P^= rr (1,5)- . 4 = 9-^ = ^'^^ P^^^^^- 
If the pressure of the atmosphere were p^ = 14,6 pounds, the tension 
of the air under the valve would be 
p = 20,26 pounds. 

§ 387. Mariotte's Lavr. — The tension of a gas increases with 
the condensation ; the more we compress a certain (juantity of air, 
the greater the tension becomes, and the more we expand or attenu- 
ate it, the less the tension becomes. The relation between the 
tension and the density or volume of gases is expressed by the law 
discovered by Mariotte (or Boyle) and named after him. It asserts, 
that the density of one and the same quantity of air is proportional to 
its tension, or, since the spaces occupied by one and the same mass 
are inversely proportional to their densities, that the volumes of one 
and the same mass of air are inversely p)roportional to their tensions. 



§387.] EQUILIBRIUM A^D PRESSURE OF THE AIR. 



781 



Fig. 657. 



D 
E 

A 



1 


iiliiiiiliilp 



;F 



B 



If a certain quantity of air is compressed into half its original 
Yolume, that is if its density doubled, its tension becomes twice as 
great as it was in the beginning, and if, on the contrary, a certain 
quantity of air is expanded to three times its original volume, its 
density is diminished to one-third of what it was, and its original 
tension is also diminished in the same proportion. If the space 
below the piston U F of sl cylinder A C, Fig. 657, is filled with 
ordinary atmospheric air, which in the beginning- 
acts with a pressure of 14,7 pounds upon each 
square inch, it will act with a pressure of 29,4 
pounds, when we move the piston to E^ F^ and 
thus compress the inclosed air into one-half its 
initial volume ; the pressure will become 3 . 14,7 
= 44,1 pounds, when the piston in passing to 
E^ Fci describes two-thirds of the entire height. 
If the area of the surface of the piston is one 
square foot, the pressure of the atmosphere against it is — 144 . 14,7 
= 2116,8 pounds ; hence, if we wish to depress the piston one-half 
the height of the cylinder, we must place upon it a gradually 
increasing weight of 2116,8 pounds, and if we wish to depress it 
two-thirds of the height of the cylinder, 2 . 2116,8 = 4233,6 pounds 
must gradually be added, etc. 

We can also prove Mariotte^s Law by pouring mercury into the 
tube Go H, which communicates with the cylindrical air vessel 
A C, Fig. 658. If we begin by cutting off a certain volume A C 
of air, of the same tension as the exterior air, by 
means of a quantity D E F H oi mercury, and 
if we then compress it by pouring in quicksilver, 
until it occupies one-half, one-quarter, etc., of its 
original volume, we will find that heights Gx H^y 
G<i Hci, etc., of the surface of the mercury in the 
tube are equal to the height of the barometer & 
multiplied by one, three, etc. Consequently, if 
we add the height corresponding to the pressure 
of the atmosphere, we find that the tension is 
double, quadruple, etc., that of the original 
volume. 

The correctness of the law of Mariotte in regard to expansion 
can easily be proved by dipping a cylindrical tube (of regular cali- 
bre) A B, Fig. 659, vertically into mercury (water) and, after 
properly closing the upper end A, expanding the enclosed volume 



Fig. 658. 




782 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 387. 



of air A E (I) by carefully drawing up the tube so that the air 
shall occupy a volume A^ E^ (II). The densities of the air in 

the spaces A E and A^ E^ are in- 
versely proportional to the heights 
A C and A^ d, and its tensions are 
directly proportional to the differ- 
ences between the height h of the 
barometer and the heights C D and 
(7i i>i of the columns D E and D^ E^ 
of mercury standing above the level 
H E of the mercury ; hence, accord- 
ing to Mariotte's law, 

A C _ l-C,D, 
A,C, ~ h- CD' 
which can be verified by observing any given immersion of the 
tube A B. 

If li and lix or fj and j»i are the tensions, y and yi the corre- 
sponding densities or heavinesses, and V and Fi the corresponding 
volumes of the same quantity of air, we have, according to the 
above law, 

:^ = 17 = -^ = ^, or F y = Fi yi and Fi ^1 




r 



F 



i?i 



y J), whence 



A 



^ y = i^ y and Fi = A ^ = 



Pi 



V. 



By means of these formulas we can reduce the density and' also 
the volume of the air of one tension to those of another. 

Remark. — It is only when the pressures are very great that variations 
from the law of Mariotte are observed. According to Regnault, when the 
volume V of atmospheric air at one meter pressure becomes the volume 
Fj, the pressure is 

i? = ^ [l - 0,0011054 /^ - 1 \ + 0,000019381 /^ - 1 V 1 meters, 



so that for . 
we have . . . 



y- 5 

p = 4,97944 



10 
9,91622 



15 

14,82484 



20 

19,71988 meters. 



Example 1) If the manometer of a blowing machine marks 3 inches, 

.30 + 3 

and the barometer stands at 30 inches, the density of the blast is — —^- = 

^ = 1,1 times as great as that of the exterior air. 

2j If a cubic foot of air, when the barometer stands at 30,05 inches, 



§ 388.] 



EQUILIBRIUM AND PRESSURE OF THE AIR. 



783 



62 425 
weiglis --^i^^TT pounds, what is its weight when the barometer stands at 34 

inches ? Its weight is 

62,425 34 



42,449 



= 0,09173 pounds. 



770 • 30,05 ~ 462,77 

3) How deep can a diving-bell (Fr. cloche a plongeur ; Ger. Taucher- 

glocke) A B G I), Fig. 660, be immersed in water, when the water is not to 

rise in it above a certain height G H= y. In the 
beginning the bell with its opening G D stands ' 
above the level of the water H i?, so that the 
whole space F is filled with air at a pressure 
equal to that of a column of water, whose height 
is = &. If afterwards the bell sinks to a 
depth G = X and a volume W of water is 
thus introduced into it, the volume of the in- 
closed air, when none is pressed back through 
the hose, becomes V — TTand the height of the 




water barometer becomes 1) + x 
T) -\- X — y V 



y 



hence 



V- W 



whence we obtain 

X = y — d + 



VI 



V— w 



y + 



wi 



V- w 



If the mean cross-section of the lower part of the bell = F^ we can put 

W =^ F y and therefore 

Fl 



X 



y 



/ Fl X 

(1 + v-ZTTy)- 



If the height of barometer = 34 feet of water, the volume of the bell F= 
100 cubic feet, the mean cross-section of the lower half i^ = 20 square 
feet, and the height, to which the water is to be admitted, is ?/ = 3 feet, the 
volume of this water is W=^ ^y = 20 . 3 = 60 cubic feet ; hence that of 

the confined air is V — W = 40 cubic feet, and its density is = — = 2|- 

times that of the exterior air, and the corresponding depth of immersion is 

60 . 34 ^ ^. ^A jy ^ 

a; = 3 -I jTT — = 3 4- 51 = 54 feet. 

40 

§ 388. Work Done by Compressed Air. — The energy stored 
hy a given quantity of air when it is compressed to a certain degree, 
as well as that restored by it when it expands again, can not be de- 
termined at once ; for the tension varies at every moment of the 
expansion or compression. We must therefore seek out a particular 
formula for the calculation of this quantity. Let us imagine a 
certain quantity of air A i^^ to be shut off in a cylinder A C, Fig. 
661, by a piston E F, and let us calculate what mechanical effect ig 



784 GENERAL PRINCIPLES OF MECHANICS. [§388. 

necessary to move the piston a certain distance E Ex — F F^. If 
the initial tension = p and the initial height of the space m the 
cylinder A E = s, and if, on the contrary, the tension after the 

space E E^ has heen described = ^i and the height 
Fig. 661. ^^ ^ ^f ^^ remaining volume of air = 5i, we have 

the proportion 

^j : ^ z= 5 : 5i, whence px= — V* 

While the piston describes a very small portion 

El E.2 — o of the space, the tension pi ikn be re- 

2 garded as constant, and the work done is = Fpi o = 

F V s o 
\ — — , F denoting the area of the piston. 

According to the theory of logarithms,* a very small quantity 
X = 1{1 i- x) = 2,3026 log, (1 + x), 
I denoting the ]N"aperian and log. the common logarithm ; conse- 
quently we can put 

Fps — =Fpslll-^—j 

^2,d026Fpslog.(l + — V 
But now 

hence the elementary work done is 

Fp s — = Fp s [I (si + a) — 1 5,]. 

Let us imagine the whole space E E^ to be composed of n parts, 
such as G, I.E., let us put E Ex =^ n g,wq will then find the work 
corresponding to all these parts by substituting in the last formula 
successively, instead of Sx, the values Si + g, Sx + 2 g, Sx -{- ^ g, . . . 
up to 5i + (n — 1) G, and instead of Sx + g, the values s, + 2 cr, 
5i + 3 (T, etc., up to Sx + n G or s, and if we add the values de- 
duced, we will obtain the whole work done while the space s — Si 
is described 



x^ x^ 



* According to the series ^ z= 1 + x + :, — - + - — - — - 4- . . • (see § 194 

1 . « 1 . « . o 

and also the Introduction to the Calculus, Art. 19) for a very small «, we 

Jbave e* = 1 + «, and therefore 

Z (1 + a;) = jc. 



1 388.] 'EQUILIBRIUM AND PRESSURE OF THE AIR. 785 



Ji = Fp s 



l{si -\- o) — I Si 

I (Si -{- 2 a) - I (si -^ a) 

l(si -^ do) - I (si -h 2a) 



I {Si + n a) — I [si + (w — 1) o] 
= Fp s [I (si + n o) — I 5i] 

= Fp s (I s — I Si) = Fp s I (— j ; 

for the first term in each line is cancelled by the second term in 
the next. 

Since — = ^ = — , we can put the work done 
Si h p "^ 

If we make the space described by the piston s — Si = x, we 

find for the mean value of the pressure on the piston, when the air 

is compressed in the ratio 

hi _ pi 

h ~ p' 

p^A=Fp'-l(^\ 
X ^ X \pj 

Putting F —\ (square foot) and s = 1 (foot), we obtain the 

following formula for the work done 

^ = ^ ^ (f ) ^ ^'^^^^ ^ ^'^- (f )• 
This formula gives the mechanical efiect necessary to transform 
a unit of volume (1 cubic foot) of air from a lower pressure or ten- 
sion ^ to a higher one j9i, and in so doing to compress the air into 

a volume of ( — 1 cubic feet. On the contrary, 

expresses the work done by the unit of volume of a gas which passes 
from a greater tension p^ to a lesser one p. 

In order to compress a quantity of air, whose volume is V and 

y 

whose tension is p^ into a volume Yx of the tension p^ = ^ p, the 

work to be done is Vp I I^V and if, on the contrary, the volume 
50 



786 GENERAL PRINCIPLES OF MECHANICS. [§388. 

Vi of the tension pi becomes a volume F, whose tension is ^ = 
^ /?i, the energy restored is 



rpi(^)^r,pj(^} 



Remark. — The mechanical effect necessary to produce moderate dif- 
ferences of tension (p^ — p), or small changes of volume (F^ —V) can be 
expressed more simply by the formula 

or more accurately by the aid of Simpson's rule, when z denotes the press- 
ure at the middle of the jDath — — ^ of the piston, by the formula 



^-^hB(^^')- 



But now 

s_s 2s 2 2p^ 



P i (s + sj s + s, X ^ P_ P+Pi 

Pi 
whence it follows that 

Example — 1) If a blowing machine changes per second 10 cubic feet 
of air, at a pressure of 28 mches, into a blast at a pressure of 30 inches, 
the work to be done in every second is 

A = 17280 . 0,4913 . 28 . I /|g) = 287711 .(115-1 16) 

= 237711 . (2,708050 - 2,639057) = 237711 . 0,068993 
= 16400,4 inch-pounds = 1366,7 foot-pounds. 
The approximate formula, given in the remark, gives for this work 



A = ^. 237711 



/30 8.2 28\ 

(28 ' ^8~ - 30) = ^^^1^'^ • ^'^1^8^ 



= 16396,9 inch-pounds = 1366,4 foot-pounds. 
2) If under the piston of a steam-engine, whose area is i^ = tt . 8^ = 
201 square inches, there is a quantity of steam 15 mches high and at a ten- 
sion of 3 atmospheres, and if this steam, in expanding, moves the piston 
forward 25 inches, the energy restored and transmitted to the piston is, if 
we assume Mariotte's law to be true for the expansion of steam, 

^ = 201 . 3 . 14,70 . 15 I (^^-^) = 132961,5 1 1 

= 132961,5 . 0,98083 = 13041 3 inch-lbs. = 10866 foot-lbs., 
and the mean force upon the piston is, when we neglect the friction and 
the opposing pressure, 



§ 389.] 



EQUILIBRIUM AND PRESSURE OF THE AIR. 



787 



P = 



130413 
25 



5217 pounds. 



§ 389. Pressure in the Different Layers of Air. — The air 

enclosed in a vessel has a different density and tension at different 
depths; for the upper layers compress those below them, upon 
which they rest ; the density and tension are the same in the same 
horizontal layer only, and both increase with the depth. In order 
to find the law of this increase of the density from above down- 
wards, or of the decrease from below upwards, we make use of a 
method similar to that employed in the foregoing paragraph. 

Let us imagine a vertical column A E, Fig. 662, whose cross- 
section A B — 1 and whose height A F = s. Putting the heavi- 
ness of the lowest layer = y and its tension = jh and 
the heaviness of the upper layer E F^— y^ and its 



Fig. 662. 




tension = p^, we have 



y^ ^ Pi 
y p' 

If o denotes the height E E^ of the layer Ey F, its 
weight, which is the decrease of the tension corre- 
sponding to (7, is 

1 . 



V 



fyypi 

Ti = - 



hence by inversion we obtain 

P 



P 



V 



G — 



y Pi 



or, as in the foregoing paragraph. 

If we substitute in it for ih successively ;:>i + v, jt?i + 2 i^, ^i + 3 v, 
etc., up to p = p^ + {n — 1) V and add the corresponding heights 
of the layers of air or values of g, we obtain, exactly as in the fore- 
going paragraph, the height of the entire column of air 



or also 



y 



= ll{^^) = .,..^ilo,(i), 



when 1) and hx denote the tensions and p and jOi the corresponding 
heights of the barometer in A and F. 

Inversely, if the height s is given, the corresponding tension 
and density of the air can be calculated. We have 



Pi 






p 

e , or y, 



P 



y e 



788 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 390, 



in which e = 2,71828 denotes the base of the Naperian system of 
logarithms. 

Remabk. — This formula is employed for the measurement of heights 
by means of the barometer, a subject which is treated in the " Ingenieur," 
page 373. If we neglect the temperature, etc., we can write as a mean value 

8 = 60346 log. {^^ feet. 

Example 1) If we have found the height of the barometer at the foot 
of a mountain to be 339 and at the top 315 lines, the height of the moun- 
tain given by these observations is 

s = 60346 log. (ff f) = 60346 . 0,031889 = 1924 feet. 

2) For the density of the air at the top of a mountain 10000 feet high,, 
we have 



7 



7 



%• 7- = m^ = 0,165711, whence 

7i /I 



= 1,465 and ^ = ^— = 0,683 ; 
7 1,465 ' ' 



Fig. 663. 



its density is therefore 68|- per cent, of that of the air at its foot. 

§ 390. Stereometer and Volumeter. — Mariotte's law finds 
a practical application in the determination of the volumes of pul- 
verent and fibrous bodies, etc., by means of the so-called stereometer 
and volumeter. 

1) Say's Stereometer. — If the glass tube CD, which is immersed 
in mercury H D R and at the same time is in communication with 

the closed vessel A B, Fig. 663, I, is. 
raised up without being drawn entirely 
out of the mercury (II), then, in conse- 
quence of the expansion of the enclosed 
air, a column C U of air enters into the 
tube and a column of mercury D ^ will 
remain behind in the tube, by the aid 
of which the diminished tension of the 
enclosed air balances the pressure of the 
atmosphere. 

Now if V^ is the volume of the space 

A B C, F, the required volume of the 

body K, which is placed in it, V the 

volume of the column of air C E,l) the 

height of the barometer and h that of 

the column of mercury D E, we have, 

according to Mariotte's law, since the same quantity of air occupies 

the volume F„ — Fj, when the tension is h, and the volume F^ — 

Fj + F, when the tension i&h — h. 




390.] 



EQUILIBRIUM AND PRESSURE OF THE AIR. 



789 



Fo - Fi + F b ' 

hence the required volume of the body is 

If we know the vokime F^, and if, when making the experi- 
ment, we draw the tube so far out of the water that the length and 
consequently the volume F of the column of air in the tube C D 
becomes a certain definite one, and if we observe also the height b 
of the barometer and that h of the column of mercury D E, we can 
calculate by means of this formula the volume F of the body K, 

2) Regnaulfs Vohometer. — If the space A B CD, Fig. 664, which 
is filled with atmospheric air and which contains also the body K, 

whose volume F is to be determined, is shut off" 
by the cock C from the exterior air, and if, by 
opening the cock E, we let out so much mercury 
from the tube D E that its level descends from 
M to iV, we can again employ (according to 
Mariotte's law) the above formula 
Vq - F _ b-h 
- V,-V,+ V b ' 

in which we denote the volume of the space 
A B C I) hj Fo, that of the mercury drawn off 
by F and the height M JV of the same by h. It 
follows, exactly as in the above case, that the 
volume of the body in A is 



Fig. 664. 




F = 



F 



In order to fill the tube D E with mercury 
again for the purpose of making a new measure- 
ment, we put that tube D E yq. communication 
with the reservoir of mercury G Jlhj turning 
the cock E. 

3) Xopj/s Vokimeter. — The pressure of the 
air enclosed in the space A B C D, Fig. 665, is 
the same as that of the exterior air, when the 
surface of the mercury D G touches the lower 
opening D of the manometer D E. If by means 
of a piston P we press the mercury into D G, 
until it rises to a certain height and its surface reaches the point 




790 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 391. 



Fig. 666. 



S, the enclosed air will be compressed and the mercury will rise a 
certain distance h in the manometer, which distance can be read off 
npon the scale. If again f\ is the volume A B CD of the air, 
Fi the required volume of the body placed in it and V the volume 
of the mercury, which has been pressed into the air-vessel, we have 
in this case 

V, - F, _ b +h 

and, therefore, the required volume of the body 

F, = F. - (^^') F. 

The constant volumes V^ and V^ are determined for each par- 
ticular instrument by filling them with mercury and weighing the 
quantity which they hold. 

§ 391. Air Pump. — (Fr. machine pneumatique ; Ger. Luft- 
pumpe.) If we raise the piston K, Fig. 666, of an air pump when 

the stop-cock is in the position (I) and 
push it down when the stop-cock is in 
position (II), it acts as an exhausting or 
rarefying 2^ui^W -»' i^? ^^ ^^^ contrary, we 
raise the piston when the stop-cock is in 
position (II) and depress it when it is in 
position (I), it acts as a com])re8sing or 
condensing pump. In the first case the 
air in the receiver A is more and more 
rarefied by the reciprocating motion of the 
piston K in the cylinder C D, and in the 
latter case it is rendered more and more 
dense. 

1) The Exhaust Pump. — If F is the 
volume of the receiver, measured to the 
cock H, Fj the clearance between H and 
the lowest position of the piston, and C the volume described by 
the piston K, which is also measured by the product F s o^ the 
surface i^of the piston and the space s described by it, the pressure 
h of the air originally contained in the receiver becomes, according 
to Mariotte's law, at the end of a single stroke of the piston 

Since upon the return of the piston the clearance remains filled 
with air at the pressure of the exterior air h, if the pressure of the 





§391.] EQUILIBRIUM AND PRESSURE OF THE AIR. 791 

air in the receiver at the end of the second stroke is denoted by h^^, 
we will have 

+ TT , TT , n + ^1 ^> whence 






_ ( V Y VV, b 



F+ F, + (7/ ^ {V+V,+ Cy ' V-^V.-h G' 
In like manner for the tension b^ at the end of the third stroke 
we find 

{V+V,+ G)h=Vb,+ V, h, and therefore 



V y v v,b vv,h 

TTT, + g) "^ {V+V.-h GY'^ (F+ Fi + (7y 



~\V-i- v,+ g) 

•^F+F, + (7 \,F+F, + (7J ^L\F+F, + 67 



F+F, +(7 ^ J F+Fi+C^' 



and from the foregoing we see that the pressure b^, after n strokes, 
will be 

+ \[vtv;tc) + (-ftf.-tc) +••• + ! J 



F+F, +(7* 

F Fi 

"If we denote ^f^ — ^ — 7:, by p and ^p^ — =9^ — 7., by (7, we will have 

b„ = p" b + (1 + ^ + jf?2 ^ . . . + jt?"-i ) q h, 
or, since the sum of the geometrical series in the parenthesis is 

«" — 1 1 — »" 
= — — — - — ^— (see Ingenieur, page 82), the required final 

tension is simply 

For n — 00 , j9" becomes = 0, and consequently the smallest possi- 
ble tension is 

i-p G + v; 

2) The Condensing Pump. If we adopt the same notations as 
for the exhaust pump, we have here for the tension of the air at 
the end of the first single stroke 

( ^ + F,) ^, = ( F + F, 4- 6^ b, whence b, = (i^A+-^) b; 

and for that b^ at the end of the second stroke 



792 GENERAL PRINCIPLES OF MECHANICS. [§391. 



(F + ro b,= Vh^ + (V, + C)b, whence 

^2- (F+ Fi)^ '^ V + V, 

In like manner the tension at the end of the third stroke ia 
found to be 

[V ■\-V,)h^Vl,-V{V,^ G) h, and therefore 

^3 = \-vTvJ ^ + LI fTT; + fTT; -^ ^J TTT. *' 

or putting 

y . y^ + G 

fTT:=^^"^^^ rTF; = ^^ 

h = [pi' + (1 + i?i + i?i') q^] b. 
In general, we have for the tension at the end of the ^th stroke 
of the piston 

h = [Pi' + (1 + 7^1 + i?i' + . . . + i?r~0 <Zi] ^? or, since 



1 + Pi +jyi' + ... +^r-' = 



.-i_^i'^-l _ 1-K 



^1-1 1 -^i' 

For n = cc y 2^1" ~ and 

1 - Pi Fi 

This is of course the greatest tension that can be produced by 
this condensing pump. 

If the clearance Fi were = 0, we would have for the exhaust 
pump q = 0, whence 

bn=p"b = [ y ^ ^ t) p ; 

and, on the contrary, for the condensing pump p^ = 1 and ^ 

1 pi 

= w, and consequently 



bn = {1 -{- n q^) b = ll -\- n -y) b. 



Example. — If the volume of the receiver of an air pump is F = 1000 
cubic inches and the clearance is 10 cubic inches, -while the volume of the 
cylinder is 300 cubic inches, the tension of the air after 20 strokes is 

1) when rarifying, since 

1000 
p = j^ = 0,76336 and 



§393.] EQUILIBRIUM AND PRESSURE OF THE AIR. 793 

0,76336- + -i-z-^^g33g- • 0,0076336J 6 

= (0,0045143 + 0,0321136) & = 0,076269 &; 
©n the contrary, 

2) when condensing, in which case 

Pi=^0 = 0,99010 and 

310 
^1 = 1010 = ^'^^^^^' 

j„ = 530 ^ ^0,9901- + -Y:z-o-;§m: ' o,30693J & 

/ , 0,18046 \ 

= (0,81954 + —09901 • 0,30693j h = 6,414 5. 

§ 392. Gay-Lussac's Law. — The Jieat or temperature of 
gases has an important influence upon their density and tension. 
The more the air enclosed in a vessel is warmed, the greater its 
tension becomes, and the more the temperature of a gas, contained 
in a vessel closed by a piston, is raised, the more it will expand and 
drive the piston before it. Gay-Lussac^s experiments, repeated 
more recently by Rudberg, Magnus and Regnault, have show^n that 
for the same density the tensions, and for the same tensions the 
volume, of one and the same quantity of air increases with the 
temperature. We can place this law by the side of that of Mariotte 
and call it Gay-Lussads Law. According to the latest researches 
the increase of the tension of a given volume of air, when heated 
from the freezing to che boiling point of water, is 0,367 times the 
original tension, or if its temperature is raised that much, the vol- 
ume of a given quantity of air is increased 36,7 per cent., when the 
tension remains constant. If the temperature is given by the cen- 
tigrade thermometer, in which the distance between the freezing 
and boiling points of water is divided into 100 degrees, the expan- 
sion for each degree is = 0,00367, and for the temperature f it is 
— 0,00367 t°, or if, on the contrary, we use Reaumur's division of 
the same space into 80 degrees, we have the expansion for each de- 
gree = 0,00459, or for a temperature of f, — 0,00459 t. 

In England and America the Fahrenheit thermometer is gene- 
rally used, in which the boiling point is 212° and the freezing 
point is 32" ; hence the increase for each degree is — 0,00204, and 
for f it is 0,00204 {t - 32). 

This ratio or coefficient of expansion 6 = 0,00367 or = 0,00204 
is strictly correct for atmospheric air alone ; its value for other 
gases is generally smaller, and it varies slightly with the tempera- 
ture for atmospheric air. 



794 GENERAL PRINCIPLES OF MECHANICS. [§392. 

If a mass of air, originally of the volume V^, is warmed from 
the freezing point to t degrees without changing its tension, its 
Yolume becomes 

F = (1 + 0,00367 t) F, = [1 + 0,00204 (t - 32°)] V„ 
and if it reaches the temperature t^, the volume becomes 

F = (1 + 0,00367 /O r„ = [1 + 0,00204 {t, - 32")] V,} 
hence the ratio of the volumes is 

V_ _ (1 + 0,00367 t) _ 1 + 0,002 04 {t - 32°) ^ 
F ~ (1 + 0,00367 ^ij "~ 1 + 0,002()4"(^i -r- 32°) ' 
on the contrary, the ratio of the densities or heavinesses is 

^ _ JF _ 1 + 0, 00367 U _ 1 + 0,00204 {U - 32°) 
yi ~ T ~ 1 + 0,00367 i ~ \ ^ 0,00204 \t - 32°)' 
or generally 

X - Zi - L±A^_i - 1 + ^ (^^1 - 32°) 

yi~ F~"l + (5?^~l + (J(^f- 32°)* 
When a change in the tension also occurs, if ;;„ is the tension at 
the freezing point, i) that at the temperature t and px that at ^i, we 
have 

F=: (1 + 0,00367 ^ F„, 

Fi = (1 + 0,00367 ^0 ^" F, 

F _ 1 + 0,00367 t 2h . 
Fi ~ 1 + 0,00367 t, ' p ' 
_y _ 1 4- 0,00367 tr p 
yi ~" 1 + 0,00367 ^^ ' pi 
y 1 + 0,00367 t, b 



, or 

as well as 



yi "" 1 + 0,00367 t ' W 
p___b_ _ 1 4- 0,00367 t y_ 

"^ ■" Ti " 1 + o,oo3W7i * 7; 

When t is given in degrees of Fahrenheit's thermometer, we must 
substitute in the latter formulas for 0,00367 t, 0,00204 {t — 32°). 

Example. — If 800 cubic feet of air, at a tension of 15 pounds and 
at a temperature of 50° Fahrenheit, are brought, by means of the blow- 
ing engine and warming apparatus of an iron furnace, to a temperature 
of 392° and to a tension of 19 lbs, its volume will be 

1 + 0,00204.(392-32) 1,734 12000 ,^^^ ,. „ , 

^^ = r + 0,00204 . -(50^:32) ' ^ ' ^^^ = i;0367 * "IT = ^^^^ ^^^^^ ^''^* 

Bemakk. — The formula 

Z_Z1-L±_^ _ 1 + <^ (^1 — 32) 
7i~ y ~ \-\- 6t~ l + (5(« — 32) 



§393.] EQUILIBRIUM AND PRESSURE OF THE AIR. 795 

can be employed for solids and for some liquids ; but for every solid we 
must substitute a different coefficient of expansion, e.g., 

Centigrade. Fahrenheit. 

for cast iron, c5 = 0,0000336 = 0,0000187, 
for glass, (5 = 0,0000258 = 0,0000148, 
for mercury, d = 0,0001802 = 0,0001001. 

§ 393. Heaviness of the Air. — By the aid of the formula 
at the end of the last paragraph, we can calculate the heaviness y 
of the air for a given temperature and tension. Eegnault, by his 
recent weighings and measurements, found the weight of a cubic 
meter of atmospheric air, at the temperature 0° of the centigrade 
thermometer and at a tension corresponding to height of 0,76 
meters of the barometer, to be = 1,2935 kilograms. Since a cubic 
foot (English) = 0,02832 cubic meters and 1 kilogram — 2,20460 
pounds English, the heaviness of air under the given conditions is 

== 2,20460 . 0,02832 . 1,2935 =-- 0,08076 pounds English. 
If the temperature is = t° centigrade, we have for the French 
measure 1,2935 , ., 

and for the English system of measures and Fahrenheit's ther- 
mometer _ 0,08076 

^ ^ 1 + 0,002047^ - 32")* 
If the tension differs from the mean tension, or if the height of the 
barometer is not 0,76 meters, but h, we have 

_ 1,2935 h _ 1,702 . h 

'^ ~ 1 + 0,00367 t ' 0,76 ~ 1 + 0,00367 t ^^^^S^^^^' 
or, since in England and America the height of the barometer is 
generally given in inches, and since 0,76 meters = 29,92 English 

inches, 

_ 0,08076 _^ _ 0,002699 h 

^ ~ rT"o;0()204 {t - 32°) • 29,92 ~ 1 + 0,00204 {tr-^^^W) 

Very often we express the tension by the pressure p upon the 

square centimeter or inch, and then we must introduce the factor 

P V 

or T— -, by doing which we obtain 
IjUooo 14,7 

1,2935 p 1,2514 p , „ 

^ = iT^;oo3677 • w = T^^ro;oo3W-t ^"^^^^^^^' '' 



7 



0,08076 p 0,005494 p 



lbs. 



1 + 0,00204 {t - 32) ' 14,7 1 + 0,00204 {t - 32) 
For the same temperature and tension, the density of steam is 
about I of that of atmospheric air ; hence for steam we have 



796 



GENERAL PRINCPLES OF MECHANICS. 



[§ 394. 



r 



0,8084 



V 



0,7821 p 



y = 



0,00367 t * 1,0336 
0,050475 



kilograms, or 



1 + 0,00367 t 

^ _ 0,003434 jt? 

1 + 0,00204 {t - 32) ■ 14,7 ~ 1+0,00204(^-32)^^^ ° 

Example — 1) What is the weight of the air contained in a cylindrical 
regulator 40 feet long and 6 feet wide, when it is at a temperature of 50'' 
and its tension is 18 pounds ? The heaviness of this air is 



7 = 



0,005494 . 18 0,098892 



= 0,09539 pounds, 



1,0367 1,0367 

and the capacity of the reservoir is 

F = TT . 3^ . 40 = 1131 cubic feet; 
hence the air enclosed in it weighs 

Vy = 0,09539 . 1181 = 107,9 pounds. 
2) A steam-engine uses per minute 500 cubic feet of steam at a temper- 
ature of 224,6° F. and at a tension of 39 inches = 0,4913 . 39 = 19,161 
pounds ; how much water is needed to produce this steam ? The heavi- 
ness of the steam is 

0,003434 . 19,161 0,06580 ^ ^,_, 

= riro;oo204Ti92;6 = -i;393- = ^'^^^^^ P°^^^' 5 

hence the weight of 500 cubic feet of steam is 
Fy = 500 . 0,04724 = 23,62 pounds. 

§ 394. Air Manometer. — From the results obtained in the 
Fig 667 ^^^^ paragraphs, the theory of the air or closed manom- 
eter can be deduced. It is composed of a barometer 
tube A B, Fig. 667, of regular calibre, the upper part 
of which is filled with air and the lower part with 
mercury, and of a cistern C E R, which also contains 
mercury and is put in communication with the gas or 
vapor. From the heights of the columns of air and 
f Wi; 1 mercury in A B, the tension can be calculated in the 
following manner. The instrument is generally so 
arranged that the mercury in the tube and in the 
cistern are upon the same level, when the tempera- 
ture of the enclosed air is ^ = 10° Cent. = 50° Fahr. 
and the tension in the space E R \^ equal to the 
mean height of the barometer h ~ 0,76 meter = 29,92 
inches. 

If, when the height of the barometer is J, a column 
of quicksilver rises from the cistern E R into the 
tube to a height ^„ and if the length A 8 of the re- 
maining column of air is = hc^, the tension of the 
latter is 






.i,i'i 




§395.] EQUILIBRIUM AND PRESSURE OF THE AIR. 797 



(fix + ^2\ . 



and, therefore, the height of the barometer of the air m E R 

b^ = hi + z = hi -h (-^-T — ') b. 

Now if a change of temperature takes place, i.e., if the tem- 
perature at the time when hi and h.2 were observed, was not as in 
the beginning = t, but = ti, we have for the tension of the column 

of d^u A S 

_ 1 + 0,00204 {ti - 32) I hi + hA 

^ ~ 1 + 0,00204 (t - 32) ' V h, I ' 
and, therefore, the required height of barometer is 

1 + 0,00204(^^1-3 2) hi + h^ , 
f)i-hi + ^^ 0^00204 {t - 32) • h, ' 

For b = 29,92 inches and t = 50° Fahr. 

b, = hi + 28,86 [1 + 0,00204 {ti - 32)] ^, 

h = hi -\- h.2 denoting the total length of the tube, measured from 
its upper end A to the surface II B of the mercury. From the 
height of the barometer b inches we obtain the pressure upon each 
square inch (English) 

^- = £& ''■ + '''' -SS^'^ «'«°^°* ('■ - 3^)] i 

= 0,4913 hi + 14,179 [1 + 0,00204 (ti - 32)] ~ lbs. 

^ ,^ 1 -h 6{ti-32) , 

P^t^^g l + d(^- 32) ^ ^^ ^' ^^^^' 

(61 — hi) ih — hi) ~ \ih b, and therefore 



By the aid of this formula we can calculate the values of the 
divisions of a scale, upon which the pressure b can be read off from 
the height of the manometer. 

Example. — If a closed manometer 35 inches long, at a temperature of 
69,8° Fahr., shows a column of air 12 inches long, the corresponding height 
of barometer is 

Jj = 35 - 13 + 38,86 (1 + 0,00304 . 37,8) f| = 13 + 38,86 . 1,07707 . f| 

= 13 + 64,76 = 77,76 inches, and the pressure on a square inch is 
p^ = 0,4913 . 77.76 = 38,30 pounds. 

§ 395. Buoyant EflFort or Upward Thrust of the Air. — 

The law of the buoyant effort of water against a body immersed in 



798 GENERAL PRINCIPLES OF MECHANICS. [§ 395; 

it, discussed in § 364, can of course be applied to bodies in the air» 
If V is the volume of the body and y the heaviness of the air, in 
which it is placed, the buoyant effort, according to this law, is 
P = V y; if the body has the apparent weight G (in the air), its 
true weight (in vacuo) is 

a^= G -}- Vy, 
If, further, y^ is the heaviness of this body, we have also 

6^j = F yi, and therefore 

C 
V = — , so that we can put 

Ti 

Q y 

Gi = G -] or Gi (yi — y) = G yi, whence it follows that 



(?, = 



Vyi — y/ 



If the body is weighed upon a scale by a weight G<i, whose 
heaviness is yo, the following equation 

G. = L^ G 

holds good ; if we divide the last two equations by each other, we 
obtain the ratio of the weights 

^ __ yi 72 - y Ts 

G^ y2 ' y, - y ^ _ x' 

or, approximatively, and generally accurately enough, 

G^ yi ys Vyi yj 

or also 

J = i + .(1_A), 

e, fij, and £2 denoting the specific gravities of the air, of the body 
weighed, and of the weight itself. 

In many cases — and — are such small fractions that they can 

£1 £2 

be neglected and the true weight can be put equal to the ap- 
parent one. 

Remark, — The law of the buoyancy of the air can be employed to de- 
termine the force, with which, and the height, to which an air-halloon 
(Fr. aerostat; Ger. Luftballon) A B, Fig. 668, will rise. If Fis the vol- 
ume of the balloon, G its total apparent weight, including the car, etc., 7, 
the heaviness of the external and 73 that of the enclosed air, we have th« 

buoyant effect 

P = Vy^ = F 78 +6^, and therefore 



395.] 



EQUILIBRIUM AND PRESSURE OF THE AIR. 



799 



the necessary volume of the balloon is 



Fig. 668. 



F 



a 




71 -72 
and the heaviness of the external air, when 
the balloon attains the greatest height, is 

G 



71=72 + 



v 



From this heaviness, by means of the 
formula 



=f'ft)=^fc)' 



s =- 



found in § 389, we can determine the great- 
est height s, to which the balloon will rise, 
by substituting for y the heaviness of the 
air at the point of beginning, which must 
be calculated according to § 393. 

Example 1. — What is the ratio of the 
true weight of dry hard wood to its appa- 
rent weight, when it is weighed by means of brass weights at a tempera- 
ture of 32° and when the height of the barometer is 29 inches. The den- 
sity of the air is, according to § 393, 

y = 0,002699 . 29 = 0,07827 pounds, that of the wood 
y^ = 0,453 . 62,425, and that of brass 
72 = 8,55 . 62,425 (see § 61), 
consequently the ratio required is 
0,07827 



G 

^2 



• (oi - i) = ' + '''''''' • ''''' = '' 



00263. 



62,425 

Thus we see that one thousand pounds of wood lose about 2f pounds 
in consequence of the buoyancy of the air. 

Example 2. — If the diameter of a spherical balloon is 30 feet and the 
heaviness of the matter with which it is filled is y^ = 0,017 pounds, and 
if the weight of the balloon with the car and load is 6^ = 500 pounds, the 
heaviness of the air at the place, where the balloon ceases to rise, is 
G Q G 3000 



7l=72 + y = 72 + ^ 



rs = 0,017 + 



= 0,017 + 0,03537 



d' ~ ' ' n 30' 

= 0.05237 pounds. 

• Now if the density of the exterior air at the starting-point is 0,0800 

pomids, we have ; (i) = ^ (^^j = o,4948, 

and if we assume the ratio of the pressure per square foot to the heaviness of 

P 
the air, i.e., - = 26210, we obtain the maximum height to which the balloon 

will rise ^^p ^ /j^\ ^ ^^^^^ ^ ^ ^ ^^gg^ ^^^^^ 

7 \7i/ 



SEVENTH SECTION. 

DYNAMICS OF FLUIDS. 



CHAPTER I. 



THE GENERAL THEORY OF THE EFFLUX OF WATER FROM 

VESSELS. 

§ 396. EfElus:. — The theory of the efflux (Fr. ecoulement ; 
Ger. Ausfluss) of fluids from vessels forms the first grand division 
of hydrodynamics. "We distinguish, in the first place, the eflSux of 
water and the efllux of air, and, in the second place, efiiux under 
constant and under variable pressure. We will begin with the 
efflux of water under constant pressure. We can regard the pres- 
sure of water as constant, when the same quantity of water enters 
the vessel as is discharged from it, or when the quantity of water 
discharged is very small, compared with the capacity of the vesseL 
The principal problem to be solved is to determine the quantity of 
water or the discharge (Fr. depense ; Ger. Wassermenge), which 
passes through a given aperture or orifice (Fr. orifice ; Ger. Oeff- 
nung) under a given pressure and in a given time. 

If the discharge per second = Q, we have the discharge in 
t seconds, when the pressure is constant, 

V=Qt. 

But if we wish to find the discharge per second, we must know 
the size of the orifice and the velocity of the effluent molecules of 
the water. To simplify our researches, we assume that the mole- 
cules flow in parallel straight lines, and, consequently, form a pris- 



397.] 



THE EFFLUX OF WATER FROM VESSELS. 



801 



matic stream, vein or jet of water (Fr. veine, courant de fluide ; 

Ger. Wasserstrahl). If F is the cross-section of the stream and v 

the velocity of the water, or that of every one of its molecules, the 

discharge Q per second forms a prism, whose base is F and whose 

height is v, and, therefore, we have 

Q — F V units of volume 
and 

G = F V y units of weight, 

y denoting the heaviness of the effluent water or liquid. 

Example — 1) If water flows through a sluice gate, the cross-section of 
which is 1,7 square feet, with a velocity of 14 feet, the discharge per 
second is 

^ = 14 . 1,7 = 23,8 cubic feet, 
and the hourly discharge is 

= 23,8 , 3600 = 85680 cubic feet. 
2) If 264 cubic feet of water are discharged in 3 minutes and 10 
seconds through an orifice, the area of which is 5 square inches, the mean 
velocity of the liquid is 

V 264 264 . 144 



V = 



Ft 



Hi-"" 



5 . 190 



= 40 feet. 



Fig. 669. 



§ 397. Velocity of Efflux. — Let us imagine a vessel A C, 
Fig. 669, which is full of water, to be provided with an orifice F, 

which is rounded upon the inside and is 
very small, compared to the surface H R oi 
the water, and let us put the head of water 
F G (Fr. charge d'eau ; Ger. Druckhohe), 
which is to be regarded as constant during 
the efflux, = hy the velocity of efflux = v, 
and the discharge per second = Q, or its 
weight — Q y. The work, which this quan- 
tity of w^ater can perform w^hile sinking 
through the distance h, \^ = Q h y, and the 
energy stored by the discharge, whose weight 
is Q y, in passing from a state of rest to the 

velocity v, is ^ Q J {% ^^)' If ^^ loss of mechanical effect takes 

place during the passage through the orifice, the quantities of work 

are equal to each other, ot h Q y = ^r— Qy,i. 

^9 




v^ 



n = 



51 



2/ 



802 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 397. 



and inversely 
in meters 



Fig. 670. 
B 



h = 0,0510 v' and^ = 4,429 Vh, 

and in feet (English), 

h = 0,0155 v' and v = 8,025 Vh. 

TJie velocity of the effluent loater is tlie same as that of a 'body 
tvhich has fallen freely through a height which is equal to the head of 
water. 

The correctness of this law can also be shown by the following 
experiment. If in the vessel A C F, Fig. 670, we make an orifice 

directed upwards, the jet F K will rise verti- 
cally and will nearly reach the level H R oi 
the water in the vessel, and we can assume 
that it would actually reach it, if all impedi- 
ments (such as the resistance of the air, the 
friction upon the sides of the vessel, the dis- 
turbance caused by the falling back of the 
water upon itself, etc.) were removed. Since 
a body which rises vertically to the height h 
has an initial velocity 

V — V2 g h, 
it follows that the velocity of efflux must be 

V =z \/2 g h. 

For another head of water hi the velocity 
of efflux is 




hence we have 

V : Vi = Vh : V hi; 
the velocities of efflux are, therefore, to each other as the square roots 
of their heads of water. {l 

Example — 1) The discharge per second through an orifice whose area 
is 10 square inches, under a head of water of 5 feet, is 



§ = i^?; = 10 . 12 V 2 ^ A=:120 . 8,025 V'5 = 963 . 2,236=2153 cubic inches. 

2) In order that 252 cubic inches of water shall pass in one second 
through an opening of 6 square inches, the head of water must be 



~ 2g~'^g \fJ ~ 



0155 



12 • V 6 / 



/252\2 0,0155 ,„, .«^. , 
I — I = ^-r^- . 42^ = 2,28 inches. 



12 



I 398.] 



THE EFFLUX OF WATER FROM VESSELS. 



803 



§ 398. Velocities of Influx and Efflux. — If the water flows 
in with a certain velocity c, we must add to the mechanical effect 

h Q y the energy - — Q y, possessed by the influent water and cor- 

Z g 



7f_ 
¥g' 



responding to the height Ih = ^, due to the velocity ; hence we 
must put 

(h ■\- h) Qy = — Qy,OYh + h,= 

and the velocity of efflux 

V = V2g{h -t Ih) = V2gh + cl 
If the vessel is maintained constantly full, the quantity of the 
influent water is equal to the discharge Q, and we can put G c = 
F V, in which G denotes the area of the cross-section R B (Fig. 

F 

669) of the water that is flowing in. Putting c = -^ v, we obtain 



h = ^ 



^g 



whence 



IFV jf_ _ - /^y 
\Gl2g~ L \g) 



2~g' 



V — 



V2 g h 



According to this formula, the velocity increases with the ratio 

F . 

^^ of the cross-sections, and it is a minimum and = y 2 a li, when 
G ^ 

the cross-section F of the orifice of discharge is very small, com- 
pared with that G of the orifice of influx, and it approaches nearer 
and nearer to infinity, the smaller the difference between the two 

F V2 a h 

orifices becomes. If i^ = 6^ or -^ = 1, we have v = — ~ — = oo, 

and also c = oo ; this infinite value must be understood 

thus : if a vessel A C, Fig. 671, is without a bottom, water 

must flow in and out with an infinitely great velocity or 

the stream of liquid G F will not fill the orifice of exit 

Gc 
CD. Putting V = ^fr^yWe obtain 




F 



h = 



(f) - '] 



^9 



, and therefore F — 



G 



D \m c 



i/ 



1 + 



2gh: 



804 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 399. 



which expression shows that the cross-section F of the discharging 
stream is always smaller for a finite Telocity of influx than that Q 

through which the water flows in, and that it therefore does not fill the 

C 
orifice of efiluXj when the latter is larger than 



/: 



1 + 



%gh 



Remabk. — The correctness of the formula 



V = 



^2 g h 



which was first established by Daniel Bernoulli, was afterwards much 
disputed. I have endeavored to prove in the " Allgemeinen Maschinen- 
encyclopadie," by Hulsse,inthe article " Efilux" (Ausfluss), how unfounded 
were the representations, which were made. 

Example. — If water flows from a vessel, whose cross-section is 60 square 
inches, through a circular orifice in the bottom 5 inches in diameter under 
a head of water of six feet, its velocity is 

8,035 V6 8,025 . 2,449 19,653 _ 19,653 



V = 



li 



i/-&;y 



Vl — (0,327)' V0,8931 



0,945 



= 20,79 feet. 



399. Velocity of Efflux, Pressure and Heaviness. — The 

formulas, which we have found, hold good so long only a^ the pres- 
sure of the air upon the surface of the water is the same as that 
upon the orifice of efflux ; but if these pressures differ, these formulas 
must have an addition made to them. If the sur- 
face H J?, Fig. 672, is pressed upon by a j^iston K 
with a force Pi, as occurs, e.g., in fire engines, we 
can imagine this force to be replaced by the pres- 
sure of a column of water. If Ai is the height 
L K Qi this column and y the heaviness of the 
liquid, we can put 

P,^ GJi, y. 

Substituting for h the head of water ^ + A] = A -f- 

p p 

-73-^, which has been increased by hi = -7^5 we ob- 
G y ^ GY 

tain for the velocity of efflux 




/ 1 p 



(^ + Wy)' 
when we assume ^ to be very small. If we denote the pressure 
upon each unit of the surface Ghj p^, we have more simply 



F 



§399.] THE EFFLUX OF WATER FROM VESSELS. 805 

and therefore 

Finally, if we denote the pressure of the water at the orifice of 
efilux by^j we can put 



P = (n + ffy, 



or 



h -\- — = -, whence 

r y 



v = y 



^9 



y 



Hence the velocity of efflux is directly proportional to the square 
root of the pressure upon the unit of surface and inversely to the 
square root of the density or heaviness of the liquid. When the 
pressure is the same, a liquid four times as heavy as another dis- 
charges one-half as fast as the latter. Since air is 770 times 
lighter than water, it would, if it were inelastic, flow out under the 

same pressure i/TTO = 27| times faster than water. 

This theory is also applicable to the case where the effluent 

water is subjected to the pressure of a column of another liquid. 

If above the level H Roi the water HER 
in a vessel A C D, Fig. 673, there is still 
a column of liquid II R^, whose height 
G .Gi — ^ and whose heaviness = y^, while 
that of the water is = y, we can replace 
the latter by a column of water whose 

height is — hi without changing the pres- 
sure upon H R or causing the velocity v 
of the water, which is passing through the 
opening F, to vary. Hence if h is the 
head F G of water, le., the height of the 

surface of separation II R above the orifice F, we have the height 

due to velocity 




v^ 



= h + -^h^, 

7 



806 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 399. 



and therefore 



v = ]/2g(h + ^h,). 



Fig. 674. 



Now if yi < y or A + — ^1 < ^ + ^1, the jet F K, which rises 

vertically, will not reach the level Hi R^ Liof the surface of the 

liquid. 

If the surface of separation H R, Fig. 674, is not above, but a 

certain distance E F = h below the 
orifice F of the vessel ADC, while the 
surface Hi Ri of the liquid Hi D R is 
at the height G Gi = hi above the sur- 
face of separation H R, we have 

f = ^hi-h, 
and therefore the velocity of the jet 




V — 



vWi^h - h). 



This supposes — Aj y h,or~y - . 

r h y, 

From this it is easy to see that the jet 
F Ky which is projected vertically up- 
wards, can rise above the surface Hi Ri of 

the liquid Hi D R. If G M = ^ hi is 

y 

the head of the liquid, reduced to that of water, M gives the level 
to which the jet will nearly reach. 

If the water does not discharge freely, but under luater, a dimi- 
nution of the velocity of efflux takes place owing to the opposite 

pressure. If the orifice F of the vessel A C, 
Fig. 675, is at a distance F G = h below the 
upper level H R of the water and at a dis- 
tance F G\ — hi below the lower level Hi Ri, 
we have the pressure from above downwards 

p =z hy, 
and the opposite pressure from below up- 
wards 

Pi = hiy; 

hence the force, which produces the efflux, is 

P ~ Pi = {h - ^h) y 



Fig. 675. 
A B 



, ,, 111, nil , -rlj I lllill 



h,;G; 



.1 III I I 
• I III II 



m 



rBi 






%:-:^. ::: — '^-s^'ir."-^ 



-ij 



§ 399.] 



THE EFFLUX OF WATER FROM VESSELS. 



807 



and the velocity of efflux is 



When water discharges under water, we must regard the differ- 
ence of level h — hi between the surfaces of water as the head of 



water. 




If the water at the orifice of efflux is pressed 
upon with a force p and at the surface or ori- 
fice of influx with a force p^, we have in general 



V 



= y2 g(h + 



Pi - P 

7 



)■ 



This case occurs when water flows from one 
closed vessel ABC into another closed one 
D E, Fig. 676. Here h is the height F G of 
the surface of the water H R above the orifice 
F, px the pressure of the air m A H R and p 
the pressure of the air or the steam in D E. 

Example — 1) If the piston of a fire engine is 12 
inches in diameter and it is pressed down in the 
cyfinder with a force of 3000 pounds, and if there are 
no resistances in the pipes and hose, the water will 
pass through the nozzle of the hose with a velocity 

= 8,025 i/- 

' 7 



-/ 



^g 



i>] 



\^%g^' 



3000 



Gy 



/ 



4 



62,5 



= 8,025 4/ 64 . - = 62,74 feet ; 



TT 



if the stream is directed vertically upwards, it will reach a height 

Ti = 0,0155 . v"" = 61,007 feet. 

2) If water flows into a space in which the air has been rarified, e.g., 
into the condenser of a steam engine, while its upper surface is pressed 
upon by the atmosphere, we must employ the last formula for the velocity 
of efflux, viz., 



, = ./,,(, + ^^). 



If the head of water is ^ = 3 feet, the height of the barometer of the exte- 
rior air 29 inches and that of the enclosed air 4 inches, w^e have 



P^ 



P 



= 29 — 4 = 25 inches = 2,083 feet of mercury 
= 13,6 . 2,083 = 28,33 feet of water. 



808 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 400. 



hence the velocity of the water flowing into the space, which is filled with 
rarefied air, is 

V = 8,035 V3 + 28,33" = 8,035 V31,33 = 44,93 feet. 
3) If the water in the feed-pipe of a steam boiler stands 13 feet above 
the level of the water in the boiler and if the pressure of the steam in the 
latter is 30 pounds and that of the exterior air is 15 pounds, the velocity 
with which the water enters the boiler is 



. = 8,035 |/l2 + ^'' - fi • ^'^ = 8,025 



/- - '-. 



144 



63,5 ' f' 63,5 

= 8,035 Vl3 - 11,53 = 5,56 feet. 

§ 400. Hydraulic or Hydrodynamic Head. — If the water 
in a vessel is in motion, it presses less against the sides of the ves- 
sel than when it is at rest. We must, therefore, distinguish the 
hydraulic or hydrody7iamic from the hydrostatic head of water. 
If jt?i is the pressure upon each unit of the surface of the water 
H^ i?i = G^, Fig. 677, /? the pressure at the orifice F and h the 
head of water F G^, we have the velocity of efflux 



^+^=[^-©1 



01 

y L \wj2^ 

now if in another section Hc^ R^ = G^, which is at a distance 

F G^^^hx above the orifice, the pressure is 
= ^2j we have in like manner 



Fig. 677. 



hr + 



i '. 1 1! '. ' '1,' .' I ','1 



7 



'Fy 
gJ J 2 g 



If we subtract these two equations from each 
other, we obtain 



R' 



'C 



»-'-'^' =[©■-©■] 






or, if we denote the head of water Gi G^ of the 
layer H^ R^ = ^2 ^J h^, we have for the hydro- 
dynamic head at H^ R^ 



Fv . 



Pi — 

y 



=»■-?- [©■-©■] 



^9 



But -j^- is the velocity v^ of the water at the upper surface G^, 

F V 
and -^- the velocity Vi of the water in the cross-section G^, we 

can, therefore, put 



§400.] 



THE EFFLUX OF WATER FROM VESSELS. 



809 



7 



= ^ + h 



P2 



\2g 2 ff)' 



The hydraulic head — at any position in the vessel is equal to 

the hydrostatic head — + A25 diminished hy the difference of the 

heights due to the velocities of the water at this point and at the inlet 
orifice. If the free surface 6^1 of the water is very great, we can 
neglect the velocity of influx and put 



P^ 

y 



Px 

7 



+ ho- 



V, 



^9' 



hence the hydraulic head is less than the hydrostatic head hy an 
amount equal to the height due to the velocity of the water. The 
quicker the water moves, the less it presses upon the sides of the 
pipe. For this reason pipes often burst or leak for the first time, 
when the motion of the water is checked, when the pipes clog, etc. 
By means of the apparatus A B C D, represented in Fig. 678, 

the difference between the hydraulic and 
the hydrostatic head can be ocularly dem- 
onstrated. If from the cross-section G^ 
we carry a tube ^ i? upwards the latter 
will fill with water, which will rise above 
the level H R of the water, when G^ > G^ or 
^2 < '^^1 ; for, since the pressure p^ upon 
the surface of the water is balanced 'by the 
pressure of the air upon the mouth of the 
tube, we can put the height, which meas- 
ures the pressure in Gc^, 



Fig. 678. 

A B 




X 



7 



= h 



(vl _ vA 
\2 g 2 gr 



Vi 



V 



and a; is > ^2 when ^r-^ < ■^. If, on the contrary, the cross-sec- 

2g 2g ^ 

tion Gz < Gx, the water flows more rapidly through Gi, and we 

have for the height of column of water in the tube E^, inserted at 6^3, 

y -^' \2g 2 gP 

which is less than h^, so that the water does not rise to the level 
H R oi Gx. If, finally, G^ is very small and the corresponding ve- 



810 



GENERAL PRINCIPLES OF MECHANICS. 



[^ 401. 



^4 



V 



Fig. 679. 
A B 



locity very great, ~ — - can be > Ih, and the corresponding 

hydraulic head z will be negative, i.e. the 
pressure of the air on the outside will be 
greater than that of the water within. 
Hence, if a tube is carried downwards and 
its end placed under water, a column of 
water Ec^ K will rise in it, which, together 
with the pressure of the water, will bal- 
ance the atmospheric pressure. If the 
tube is short, the water in the vessel A", 
which, in this experiment, should be col- 
ored, will rise in the tube, enter the reser- 
voir A B CD and flow, with the other 
water, out at F. 

Remakk. — If the vessel A G E^ Fig. 680, consists of a reservoir A C and 
of a narrow vertical tube C E^ the hydrodynamic head is 
negative in all parts of this tube. If we do not regard the 
pressure of the atmosphere p^, the pressure of the water at 
the orifice of efflux is = ; for litre tlie entire head of water 




Fig. 680. 



A 






■■',11, i'/ 



E, 



E 



a 



¥,. 



D, 



[4D 



is expended in producing the velocity d = \J '^ gJi-^ on the 
contrary, for a position D E^ which is at a distance G^ G = 
Aj under the water level, the hydraulic head is 

= h^ — 7i = — {h — h^), 

or negative ; if, then, a hole were bored in this tube, no water 
would escape, but, on the contrary, air would be sucked in 
and discharged at F. This negative pressure is a maximum 
directly under the reservoir, since h^ is here a minimum. 
§ 401. Rectangular Lateral Orifices. — By the aid of the 
formula 

Q = Fv = FV^gh, 
the discharge per second can be calculated only when the orifice is 
horizontal, since in that case the velocity is uniform in the whole 
cross-section F\ but if the cross-section is inclined to the horizon, 
if, E.G., the opening is in the side of the vessel, the molecules of 
water at different depths flow out with different velocities, and the 
discharge can no longer be regarded as a prism ; hence the formula 

Q — Fv = FV'ilgh cannot be applied directly. The general for- 
mula is 

Q = F, VYfh, + F, V2 g li, + F^ V¥Jh, -f- . . . 

= VJ~g (F, VT, + F, VT, + i^3 ^^3 + . . '), 



§ 401.] 



THE EFFLUX OF WATER FROM VESSELS. 



811 



Fig. 681. 



in which F^, F^, i^3 • • • denote the areas and ^i, As? A3 . . . the heads 

of water of the various portions of the orifice. 

The simplest case is that of efSux through a notch in the side, 

weir or overfall, Fig. 681. The notch D E G H \n the wall, through 

which the efflux takes place, is rec- 
tangular; let us denote its width 
D E =^ G Hhj I) Sind its height 
D H — E G \>j h. If we decom- 
pose this surface h li, by horizontal 
lines, into a great number n of hor- 
izontal strips of equal width, we can 
consider the Telocity to be constant 
for each of them. Since, if we pro- 
ceed from above downwards, the 
heads of water of these strips are 
h 




n 



n n 



we have for the corresponding ve- 
locities 



/ 3 ^ . ^, l/iT^y 2 </ . '-^, etc., 



and since the area of each of these strips is = 5 . - = — , we have 

■^ n n 

the corresponding discharges 

h hh^ 



n 



/ 



%g. 



n n 



%g. 



%h Ih ./ ^ 



n 



n 



3 h 

n ' 



etc.; 



hence that of the whole section is 



Since (as is given in the Ingenieur, page 88) 



or 



1^ 



+ 2* + 3* + . . . + ni = --^-^ = I ^f = I 7^ VT, 



it follows that the required discharge is 



812 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 401. 



ti h 4/9 a h 

If we understand hy the mean velocity v that velocity ^ which must 
exist at all points of the overfall, tvhen the sams quantity of water 
passes through the whole cross-section with a uniform velocity as does 
pass through loith the variable velocity, we can put 
Q = J) Ji V, whence it follows that 

V = 'j V2 g h, 
I.E. the mean velocity of water flowing out through a rectangular 
notch m the side of a vessel is | the velocity at the sill or lower edge 
of the notch. 

If the rectangular orifice K G, Fig. 682, with the horizontal 

base G H, does not reach to the level 
of the water, we find the discharge 
through it by regarding it as the dif- 
ference between two notches in the 
/ side I) U GIIsLudD U LK. If A, 
is the depth E G oi the lower and 
h^= E L that of the upper edge, we 
have for the discharges through these 
notches 



Fig. 682. 




il)V¥gh;% 



and 



'2 J 



l^V^gh, 
hence the discharge through the rec' 



tangular opening G H K Lh 

Q = lh V2jh^' - P V2JV = P ^ (Ai^ - h,l), 
and the mean velocity of efflux is 



^ = V- 



Q 



= ihV2g . 



hi^ — ^2 



b (Ai — As) 
If h is the mean head of water E M = 



hi — ^2 
Ai 4- ^2 

2 



, or the depth of 



the centre of the orifice below the level of the water, and a the 
height K H^=^ L G = h^ — A2 of the orifice, we can put 



v^^lVTg. 



(^ - tf- (^ - i) 



a 



, or approximatively 



[i--(l)>^^*- 



§402.] THE EFFLUX OF WATER FROM VESSELS. 813 

Example. — If a rectangular orifice of efl3.ux is 3 feet wide and 1^ feet 
high and the lower edge is 2f feet below the level of the water, the dis- 
charge is 

Q = f . 8,025 . 3 (2,75! - 1,51) = 16,05 (4,560 - 1,837) 
= 16,05 . 2,723 = 43,7 cubic feet. 
According to the approximate formula 

«, = [l - ^ (^^)1 • ^'^^^ ^^^ = (1 - 0,0036) 11,698 
= 11,698 — 0,042 = 11,656 feet, 
and the discharge is, therefore, 

^ = 3 . f . 11,656 = 43,710 cubic feet. 

Remark. — If the notch in the wall is inclined to the horizon at an 

angle 6, we must substitute for the height of the orifice -^. — j^ instead of 
Aj — Ag, and therefore we must put 

e = I ^;!^ (Vv - vv). 

nn. 6 ^ ^ ^ 

If the cross-section of the reservoir, from which the water is dis- 
charging, is not much larger than the cross-section of the orifice, we must 

F 

take into account the velocity of approach ■Wj = ^ ■« of the water and put 

§ 402. Triangular Lateral Orifice. — Besides rectangular lat- 
eral orifices, triangular and circular ones also occur in practice. 
We will next discuss the discharge through a triangular orifice 
D E G, Fig. 683, with a horizontal base E G and with its apex D 
at the level of the water H R. If we put the base 
Fig. 683. ^ 6^ = J and the height i) ^ = A and if we divide 
the latter into n equal parts and pass through these 
divisions lines parallel to the base, we divide the 
entire surface into small strips, whose areas are 

,bh2bh3bh. ^ , , , „ 

-*-.-, — . -, — . -, etc., and whose heads of 
n n n n n n 

water are -, — , — , etc. The discharges through them are 
71/ it n 



] 



'/ --, 



t 



^y^^n^^^y^^i^'^^^^i^''^''^ 

by summing these we obtain the discharge of the whole triangular 
orifice 



814 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 402. 



Q 









= --^r— (It + 21 + 3f + . . . + n% 

T^ \ n 

(^S + 1 \ 
— j= I rihy 

q^llh VYgh = f:V~2gh\ 
If the base D K of the orifice D G K lies in the surface of the 
water and the apex G is at the depth Ti below it, we have the cor- 
responding discharge, since that through the rectangle D E G K 



Q^ihJi VYJh-^bh V2gh= j%ih V2gh. 
The discharge through a trapezium A B G D, Fig. 684, whose 
upper base A B = hi lies in the surface of the water, whose lower 
base is C D = I).2 and whose height is D U = h, is found by com- 
bining the discharge through a rectangle with those through two 
triangles, and it is 



^j%{2bi 4- Sb,)hV2gh. 
Fig. 684. 
HA E F BR 




Fig. 685. 
HA ^ BR 




Further, the discharge through a triangle CLE, Fig. 685, 
whose base is D E = t^, whose altitude is M — h^ and whose 
apex C is situated at a depth G = h below the level II R of the 
water, is Q = discharge through ABC minus that through A E 

= A 5 A vYjh - -2. (2 5 + 3 1,) h V 27a; 

= /_ ^TYg [2 h (M - All) -Sh^ h^]. 
Since the width A B = b is determined by the proportion 
1) : ii : : h : (h — Aj), it follows that 

2VYg. I, (2 A (Al - All) 



Fig. 686. 



Q 




_ 2 \^ 2g ,d, / 2At-5AAil+ 3 A A 



15 \ A - 7ii 

Finally, we have for the discharge through a 
triangle AGE, Fig. 686, whose apex lies above 
its base, 



§ 403.] 



THE EFFLUX OF WATER FROM VESSELS. 



815 



2V2g . h /2M-6hh,^ + 3 h^ 



15 

2h^ 



H 



k — hi 



) 



)• 



__ 2V2g.br / SB -5 h, h% 

~ 15 I h-h. 

Example. — What is the discharge through the square orifice AB C D^ 
Fig. 687, whose vertical diagonal A C =^1 foot, when the comer A reaches 
to the level of the water ? The discharge through the upper half of the 
square is 

§ = f & V2y^^ = f . 1 . 8,025 Vf = 1,605 . 0,7071 = 1,135 cubic feet, 
and that through the lower half 

2 l ^¥g /2M — 5hh^^ + d \i 




Qi = 



2 



15 

8,025 



15 
32,10 



5 /2 



h — h^ 

5 (|-)f + 3 (0 
1-i 



■) 



15 



(2 - 1,7678 + 0,5303) 



= 2,14 . 0,7625 = 1,632 cubic feet, 
consequently the total discharge is 

Q = 1,135 + 1,632 ^ 2,767 cubic feet. 

§ 403. Circular Lateral Orifices. — The discbarge for a cir- 
cular aperture A B, Fig. 688, can only be determined by means 

of approximate formulas obtained in the follow- 
ing manner. Let ns decompose the circular ori- 
'ji fice by concentric circles into small rings of 
equal width and let us consider each ring to be 
composed of elements, which may be regarded as 
parallelograms. If r is the radius, h the width 
and n the number of elements of one of these 

2 TT r . 
rings, is the length of one of these elements 




n 



and its area is 



2 IT r h 

n 



Now if h is the depth C G of the centre C below the level of the 
water II R and the angle A C K^ which measures the distance of 
the element K from the highest point A of the ring, we have for 
the head of water of this element 

KN= G - C L^h- r cos. tp, 
and therefore the discharge through this element 

2 n r b 



n 



V2 g {h — r cos. (p). 



816 GENERA.L PRINCIPLES OF MECHANICS. {§403. 

But 

V h — r COS. <f) 

= iHi jl — i ^ cos, — H (-T7 ^^^^ + . . .1 

= |/~^ [l ~ i ^ COS. </) — jV l^ (1 -f COS. 2 0) + . . .l 
and therefore the discharge through this element is 

= ^-^ ^^ \}~''\ '"'• ^ " ''^ ( D^ ^^ + '"'• ^ '^) + '• •} 

The discharge through the whole ring is found by substituting 
in the parenthesis instead of 1, ^ . 1 = m, and instead of cos. the 
sum of all the cosines of from = to </) = 2 tt, and instead of 
COS. 2 the sum of all the cosines Qii% ^ from 20 = to 20 = 4 tt. 
Since the sum of all the cosines of a full circle is equal to 0, these 
cosines disappear, and we have the discharge through the ring 



^nrhV'^gh [l - yV (X) - ...} 



T . T 2/ T 3 T 

If. instead of 5, we substitute — , and mstead of r, — , — , — to 

m m m m 

— , we obtain the discharge through each of the rings, which form 

the entire circle, and finally the discharge through the entire circu- 
lar aperture is 

e=2^r^3p(^(l + 2 + 3+...+m)-J,^(r+3' + 3'+...+m')) 

= ,7 r^ VWJl [l - ^\ (0'- . . .] 
or more exactly 

e = ^ r' V2ih [l -,.3(0'- „^j (0*- • • •] 

If the circle reaches to the level of the water, we have 
Q = tW? ^ ^' V%gh = 0,964 F Vfjh, 
when F = 7T r^ denotes the area of the circle. 

Moreover, it is easy to understand that in all cases, where the 
head of water at the centre is equal to or greater than the diameter 
of the orifice, we can put the value of the entire series = 1 and 

Q = FV2~pi. 
This rule can also be applied to other orifices and also to all 



§404.] THE EFFLUX OF WATER FROM VESSELS. 817 

cases, where the depth of the centre of gravity of the orifice below 

the level of the water is as great as the height of the aperture ; we 

can then regard the depth h of this point as the head of water and 

put Q =F V¥Yh. 

If we consider that the mean of all the cosines of the first 

2 
quadrant is = - and that of all those of the second quadrant is 

2 
— , or that the mean of the first and second quadrant = 0, the 

IT 

discharge for the wpi^er semicircle, determined in the manner 
shown above, is 

and that through the loioer semicircle is 

=^v.Ti[i+3^(9--3'.(,:)v...} 

in which F denotes the area of the aperture. 

The formulas for Q, Qx and Q^ hold good also for elliptical 
orifices with horizontal axes ; for the discharges, when the other 
circumstances are the same, are proportional to the widths of the 
apertures and the width of an ellipse is proportional to the width 
of an equally high circle (see Introduction to the Calculus, Art. 12). 

Example. — What is the hourly discharge through a circular orifice 1 
inch in diameter, when the level of the water is one fine above the top of it ? 



Here we have 

r 



= f; hence (^y^ ft =0,735, 

and 1 - ^ Q J = 1 - 0,023 = 0,977, 

and consequently the discbarge per second is 

q = '^i^ . 12 . 8,025 i/r^ . 0,0977 = ^ . 8,035 . 0,977 V7= 16,29 c inches, 

4 ' r 144 4 ' ' 

, per minute = 977,4 cubic inches, and per hour = 33,94 cubic feet. 
^' § 404. EfEIux from a Vessel in Motion. — The velocity of 
efflux changes when a vessel, originally at rest or moving uni- 
formly, is set in motion, or when a change in its condition of 
motion takes place, since in this case every molecule of the water 
acts upon those surrounding it not only by its weight, but also 
by its inertia. 52 



818 



GENERAL PRINCIPLES OF MECHAJNICS. 



[§ 404. 



Fig. 689. 




If the vessel A G, Fig. 689, is moved with an accelerated motion 
vertically upicards, while the water flows through an opening F in 

the bottom, the velocity of efflux is 
augmented, and if it descends with an 
accelerated motion, the velocity is dimin- 
ished. If the acceleration is j^^ every 
molecule M of the water presses not 
only with its weight M g, but also with 
its inertia M p, and in the first case we 
must put the force of each molecule 
equal to (g + p) if, and in the second 
case equal to {g — j)) M, or instead of 
g, g ± P' Hence it follows that 

— = {g±p)h, 

and that the velocity of efflux is 

V = V2 {g ± p) h. 
If the vessel rises with the velocity g, we have 

V =: V2 .2g h = 2 Vgli, 
and the velocity of efflux is 1,414 times as great as it would be if 
the vessel stood still. If the vessel falls by its own weight or 

with the acceleration g, ?; is = 1^0 == and no water runs out. If 
the vessel moves uniformly upwards or downwards, v remains = 

V2 g A, but if its rise is retarded, v becomes = V2 {g — p) h, and if 
its fall is retarded, v is == V2 {g -h p) h. 

If the vessel, from which the water flows, 
is moved horizontally or at an acute angle 
to the horizon, the surface (see § 354) be- 
comes oblique to the horizon and a varia- 
tion of the velocity of efflux is the result. 
If a vessel A (7, Fig. 690, is caused t6 

revolve ahout its vertical axis X X, its sur- 
face will assume, according to § 354, the 
shape of a parabolic funnel A B, and at 
the centre M of the bottom the head of 
water Jf is smaller than near the edge, 
and the water will flow more slowly through 
an orifice at the centre than through any 
other equally large aperture in the bottom. 
If h denotes the head of water M at the centre i¥, the velocity 



Fig. 690. 
X 




§ 404] 



THE EFFLUX OF WATER FROM VESSELS. 



.819 



of efflux through an aperture at that point will he — V2 g h ^ but 
if y denotes the distance M F = JSf F of an aperture F from the 
axis X X and w the angular velocity, we have, since the subtan- 
gent TX oit\\Q arc OP of the parabola is equal to twice the abscissa 
X, the corresponding elevation of the water above the centre 
X= ^, T X = I F X. tang. X F T, 

consequently if we substitute tang. X F T = tang, (p = — - (see 

§ 354) and denote the angular velocity o) h of F by w, we can put 

y _ ^ y __ w" 



X= X = Uj 



(x) 



(J ^g ^ 

Hence the velocity of efflux through the orifice F is 



V 



gh 



w 



Fig. 691. 
X 



h£i' .tiimiiiM i iiiitiiLi;iiimiiiiiimiiiiii| ||*j 

E 




This formula holds good for a 
vessel of any shape, even when it 
is closed on top, like A C, Pig. 691, 
in such a manner that the fun- 
nel DOC cannot be completely 
formed. Here also li is the depth 
M of the orifice below the vertex 
of the funnel and v the velocity 
of rotation of the aperture. It will 
be employed repeatedly in the dis- 
cussion of reaction wheels and tur- 
bines in another part of the work. 

Example — 1) If the vessel A C, Fig. 689, which when filled with water 
weighs 350 pounds, is drawn upwards by a weight G of 450 pounds by 
means of a cord passing over a pulley, it rises with an acceleration 
450 — 350 100 



9 = 



800 



9 = iff, 



^ 450 + 350 
and the velocity of efiiux is 

V = V2 (.</ + p) h = V2 .^gh = V| g h. 
Now if the head of water were h = 4: feet, the velocity of efiiux would be 

v = V977= 3 V32',2 = 17,02 feet. 
2) If the vessel A C, Fig, 691, which is filled with water, makes 100 
revolutions per minute and if the orifice ^ is 2 feet below the level of the 
water at the centre and at a distance from the axis X X, = 3 feet, the 
velocity of efflux is 

t> = V2 gh + w"" = y 64:A 



2 + 



'3 . - 



30 



looy 



V'128,8 + 100 . 77^ 



= Vl28,8 + 987 = Vl 115,8 = 33,4 feet. 
If the vessel stands still, we have v = Vl28,8 = 11,35 feet. 



820 



GENERAL PRINCIPLES OF MECHANICS. 



£§ 405. 



CHAPTER II. 

OF THE CONTRACTION OF THE VEIN OR JET OF WATER WHEN 
ISSUING FROM AN ORIFICE IN A THIN PLATE. 



Fig. 693. 



§ 405. CoefEcient of Velocity. — The laws of efflux, deduced 
in the last chapter, coincide almost exactly with the results ob- 
tained in practice, so long as the head of water is not very small, 
compared to the width of the aperture, if the orifice of efflux is 
gradually widened inwards and joins bottom or sides without 
forming an angle or edge. The experiments made with polished 
metal mouth -pieces by MicheloUi, Eyteliuein and others, and also 
by the author, have shown that the real effective discharge is from 
96 to 99 per cent, of the theoretical one. The mouth-piece A D, 
Fig. 692, which is represented in one-half its natural size, gave 

under a pressure of 10 feet 98 per cent., 
under a pressure of 5 feet 97 per cent., and 
under a pressure of 1 foot 96 per cent, of 
the discharge calculated theoretically (Ex- 
periments with large orifices, see Unter- 
suchungen in dem Gebiete der Mechanik 
und Hydraulik, Zweite Abtheil.). If the 
efflux through such a mouth-piece is to be 
as free from disturbance as possible, the 
rounding must not be in the form of a 
circle, but in that of a curve A D = B G, 
the curvature of which gradually decreases from within outwards 
(from A towards D). Since in this case the stream has the same 
cross-section F as the orifice, we can assume that the diminution 
of the discharge is caused by a loss of velocity arising from the 
friction of the water upon, or its adhesion to, the inner surface of 
the mouth-piece and from the viscosity of the water. Hereafter 
we will call the ratio of the real or effective velocity to the theo- 
retical velocity v = V2 gh the coefficient of velocity (Fr. coefficient 
de Vitesse ; Ger. Geschwindigkeitscoefficient) and we will denote 
it by <p. Thus the effective velocity of efflux in the simplest case is 

Vi = (f) V = (p V2 g hy 




^ 403.] CONTRACTION OF THE VEIN OF JET OF WATER. 821 

and the effective discharge is 

Q = Fv^ = (p Fv :=z <!> F V2gk. 

Substituting for (p its mean value 0,975, we obtain (in English 
feet) 

Q := 0,975 .FVYgli = 0,975 . 8,025 F Vh ^^ 7,824 F Vh. 
The Yis viva of a quantity Q of water, issuing with the velocity 

Vi, is -— . Vi', by virtue of which it can perform the mechanical 

v^ 
effect Q y . — -. But since the weight $ y in descending from the 
Z g 

v^ v^ 

height li = - — performs the work Q y . h= Q y ^r— , it follows that 
Z g Z g 

the loss of mechanical effect of the water during the efflux is 

I.E., 



V' 



L — 0,049 . ^r— , or 4,9 per cent. 
Z g 

The water, which issues from the vessel, will therefore perform 
4,9 per cent, less work by virtue of its vis viva than by virtue of its 
weight, when falling from the height h. 

Remakk. — The author has tested the law of efflux, expressed by the 

formula 'O = V2 gli, under very different heads, viz., from the very great 
head of 100 meters to the very small one of 0,02 meters. A well rounded 
mouth-piece 1 centimeter wide gave for the heads 



li = 0,02 meters . . . 


0,50 meters 


3,5 meters 


17 meters 


103 meters 


<l> = 0,959 


0,967 


0,975 


0,994 


0,994 



See Civilingenieur, New Series, Vol. 5, first and second numbers. 

§ 406. Coefficient of Contraction. — If the water issues from 
an orifice in a thin plate (Fr. orifice en mince parol ; Ger. Miin- 
dung in der diinnen Wand), and if the other circumstances are the 
same, a considerable diminution in the discharge takes place. This 
diminution is due to the fact that the directions of the molecules 
of the water, which are passing through the orifice, converge and 
produce a contracted stream or vein (Fr. veine contractee ; Ger. 
contrahirter Wasserstrahl). The measurements of the stream, made 



822 



GENERAL PRINCIPLES OF MECHANICS. 



[{ 



403. 



by several experimenters and more recently by the author himself, 
have shown that the stream, at a distance from the orifice equal to 
half its width, experiences its maximum contraction, and that its 
thickness is 0,8 of the diameter of the orifice. If F^ is the cross- 
section of the contracted vein and F that of the orifice, we have 
therefore 

F = 0,8^ F = 0,64 F. 

F. 
The ratio -^ of these cross- sections is called the coefficient of 

contraction (Fr. coefficient de contraction ; Ger. Contractionscoeffi- 
cient j, and is denoted by a ; from what precedes we see that its 
mean value for the efflux of water through an orifice in a thin 
plate is a = 0,64. 

So long as we have no more accurate knowledge of the law of 
the contraction of the stream, we can assume that the stream flow- 
ing through a circular orifice A A, Fig. 693, forms a solid of rota- 
tion A F F A, whose surface is generated by the revolution of the 
arc ^ ^ of a circle about the axis C D of the stream. Putting^ the 

diameter A A of the orifice = d 
and the distance C D oi the con- 
tracted section E E from the orifice 
= -^ d, we obtain the radius 

MA = MF = r 
of the generating arc A Ehj means 
of the equation 



Fig. 693. 



ILr-rl 




or 






:i-^M 



A N'' = F N {% M F - E N) 
iV d (^ d\ 



4 10 \ ' ■ lOy 
from which we obtain 
r = 1,3 d. 
The velocity of efflux through orifices of this kind is about 

V, = 0,97 V. 

The contraction of the stream of water owes its origin to the 
fact that uot only the water immediately above the orifice flows 
out, but also that the water all around flows in and is discharged 
with it. The filaments of water begin to converge within the 
vessel, as is shown in the figure, and the contraction of the stream 
is caused by the prolongation of this convergence. We can con- 
vince ourselves of this fact by employing a glass vessel and putting 
into the water small bodies, such as saw-dust, bits of sealing-wax. 



§ 407.] CONTRACTION OF THE VEIN OR JET OF WATER. 



823 



etc., of nearly the same specific gravity as the water, and allowing 
them to flow out with it. 

§ 407. Contracted Vein of Water. — If the water flows 
through triangular, quadrangular, etc., orifices in a thin plate, the 
stream assumes particular forms. The most striking phenomenon 
is the inversion of the stream or the change in position of its cross- 
section in reference to the cross-section of the orifice, in conse- 
quence of which a corner of the former cross-section comes into 
the same position as the middle of one of the sides of the orifice. 
Thus the cross-section of the stream, issuing from a triangular ori- 
fice A B C, Fig. 694, is, at a certain distance from the latter, a 
three-pointed star D E F\ that from a square orifice A B C D, 
Fig. 695, is a four-pointed star E F Q H\ that from a pentagonal 

Fig. 694. 






Fig. 697. 



orifice A B C D E, Fig. 696, is a five-pointed star E G H K L, etc. 
The cross-sections are very different at diflerent distances from the 
orifice ; they decrease for a certain distance and then increase again, 
etc. ; the stream consists, therefore, of ribs of variable width and 
forms, as can be best observed when the pressure is very great, 
bulges and nodes similar in form to the cactus plant. If the ori- 
fice A B C D, Fig. 697, is rectangular, the cross-section at a small 

distance from the aperture forms also 
a star or cross, but at a greater dis- 
tance it assumes more the form of an 
inverted rectangle E F. 

Bidone observed the discharge 
from various kinds of orifices ; Pon- 
celet and Lesbros have made the 
only accurate measurements of the 
stream issuing from square orifices 
(see the Allgemeine Maschinenency 
klopadie, article "Ausfluss"). The 
last measurements have led to a small 




coefficient of 



contraction 0,503. 



824 GENERAL PRINCIPLES OF MECHANICS. [§408. 

Measurements of the water discharged through smaller openings 
have given greater coefficients of contraction ; they indicate that 
the coefficients are greater for oblong rectangles than for rectangles, 
which approach the square in form. 

§ 408. Coefficient of Efflux. — If the effective velocity of 
water issuing from an opening in a thin plate was equal to the 

theoretical v = V2g h, we would have for the effective discharge 

Q — a Fv = a F V^JTi, 
a F denoting the cross-section of the stream at the point of maxi- 
mum contraction, where the molecules of water move in parallel 
lines ; but this is by no means true. It appears, from experiment, 

that Q is smaller than a F V%g h and that we must multiply the 

theoretical discharge F V2 g h hj si, coefficient smaller than the co- 
efficient of contraction, in order to obtain the real discharge. We 
must therefore assume that, when water issues from an orifice in a 
thin plate, a certain loss of velocity takes place, and consequently 
a coefficient of velocity (/> must also be introduced ; hence the effec- 
tive velocity of efflux is 

Vi = (j) V = (f) V2 g h. 

The effective discharge is 

Q, = Fi.Vi = aF.(j)v = a(l)Fv = a(j)F V2 g h. 

Let us call the ratio of the real discharge Qi to the theoretical or 
hypothetical discharge Q the coefficient of efflux (Fr. coefficient de 
depense; Ger. Ausflusscoefficient) and let us denote it hereafter 
by jLi ; then we have 

Q,=fiQ = [iFv = fMF V¥g7i, 

and therefore 

ft =2 a <p, 

I.E. the coefficient of efflux is the product of the coefficient of velocity 
and the coefficient of contraction. 

Repeated observations, and particularly the measurements of 
the author, have led to the conclusion that the coefficient of efflux 
IS not constant for all orifices in a thin plate, that it is greater 
for small orifices and small velocities of efflux than for large 
orifices and great velocities and that it is much greater for long, 
narrow orifices than for those whose forms are regular or circular. 

For square orifices, whose areas are from 1 to 9 square inches, 
under a head of from 7 to 21 feet, according to the experiments of 



§ 409.] CONTRACTION OF THE VEIN OR JET OF WATER. 825 

Bossut and Miclielotti, the mean coefficient of efflux is ft = 0,610 ; 
for circular orifices from -h to 6 inches in diameter and under a head 
of from 4 to 21 feet, it is /^ = 0,615 or about /g. The values, which 
were obtained by Bossut and Michelotti from their observations, 
differ materially from each other ; but they do not appear to de- 
pend upon the size of the orifice or upon the head. According to 
the experiments of the author, under a head of 0,6 meters, the co- 
efficient of efflux is for a circular orifice 

1 centimeter in diameter . . . . . . , fi = 0,628 

2 centimeters " = 0,621 

3 " " =0,614 

4 " « =0,607. 

On the contrary, under a head of 0,25 meters, with the same orifice, 

1 centimeter in diameter, he found . . . . jii = 0,637 

2 centimeters " " . . . . = 0,629 

3 " " " . . . . =0,622 

4 « « « .... =0,614 

We see from these results of experiment that the coefficient of 
efflux increases when the size of the orifice and the head of water 
diminish. If we assume as mean values fi = 0,62 and a = 0,64, 
we obtain the coefficient of velocity for efflux through an orifice in 
a thin plate ^ ^ ^ ^ ^^^^^ 

KM 

or about the same as for efflux through mouth-pieces rounded in- 
ternally. 

Remakk — 1) Experiments made by Buff (see Poggendorff 's Annalen, 
Vol. XL VI) show that the coefficients of velocity for small orifices and 
small heads or velocities are considerably greater than for large or medium 
orifices and velocities. An orifice of 2,084 lines in diameter gave, under a 
head of 1| inches, fi = 0,692 and, under a head of 35 inches, fj. = 0,644. 
On the contrary, an orifice 4,848 fines wide, under a head of 4|^ inches, 
gave fj. = 0,682 and, under a head of 29 inches, fi = 0,653. The author 
also obtained similar results. 

2) For efflux under water, according to the experiments of the author, 
the coefficients of velocity are nearly 1^ per cent, smaller than for efflux 
into the air. 

§ 409. Experiments. — The coefficient of efflux fi correspond- 
ing to a certain mouth-piece can be determined, when we know the 
discharge V, which passes through the known cross-section F of 
the orifice under a head of water y^ in a certain time t ; here we 
have 



826 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 409. 



and inversely 



F= II FVTgh.t, 
V 



Fig. 698. 



In order to find its two factors, yiz. : the coefiicient of contrac- 
tion and that of yelocity, it is necessary to measure either the cross- 
section J^i = a F oi the stream or to determine the velocity of 

efflux ?'i = <p V = (j) V2 g h by means of the range of the jet. 

]^either measurement can be made with sufficient accuracy unless 

the stream is thin and the cross-section is circular. 

The circular cross-section Fi of a jet can be determined very 

simply by means of the apparatus represented in Fig. 698. It is 

composed of a ring and four sharp-pointed set- 
screws A, B, C, D, which screw in towards each 
other. The screws are directed towards the 
centre of the cross-section of the stream and are 
turned until their points touch its surface ; the 
ring is then removed from the stream and the 
distance between the opposite points of the 
screws is measured ; the mean d^ of these two 
distances is assumed to be the diameter of the 

stream. Now if d is the diameter of the cross-section of the orifice, 

we have F^ I d, V 

" = :p = 1^)' 

and therefore jw 




0--- 



a 



If we measure the range B (7 = J of a jet ^ B, Fig. 699, which 
issues horizontally from the mouth-piece S A, which is at a certain 
height A C — a above the ground, we have, according to § 36, the 
velocity of efflux 

Fig. 699. 




§ 409.] CONTRACTION OF THE VEIN OR JET OF WATER. 827 

and since v^ = (f) v = cf) V2 g h, we obtain 



<P 



^^/J! 



V 4:ah 2Vah 



wlience „ 2Vah 

a = -~ — — -. 

The determination of v is more certain when, instead of a and 
b, we measure the horizontal and yertical co-ordinates of three 
points of the parabolic axis of the stream; for the axis of the 
mouth-piece may have an unknown inclination to the horizon. 
The most simple method of proceeding is to stretch a horizontal 
thread D F above the stream and to hang three plumb-lines from 
three points D, E, and F, which are at equal distances from each 
other ; we then measure the distances D G, E H, and F K oi the 
axis of the stream from D F. li D F ^^ x is the horizontal dis- 
tance of the extreme points from each other, if the vertical dis- 
tances D G, E H, and F K — z, z^, and z^, and if we take G as the 
origin of co-ordinates, we have the co-ordinates for the point H 

x,^GL = DE=^DF=^Q.ndiy,=LH=EH~ D G = z,-z, 

and for the point IT 

x,= GM= I)F= xajidii/,=: MX = FX- D G = z,-z. 

According to § 39, if a denotes the angle of inclination of the 
axis of the stream at G, 

(/ X ^ 

yi = X, tang, a + ^— 4-^V-, and also 

ct x^ 
y^ =z x<i tang, a + - — f- \ -, or 
^ ' ^ 2v^ cos, a/ 

Cf X ^ 

y, — X, tang, a = _^-l^-^, and 



f/a — X2 tang, a 



2 Vi COS. a 
g x^ 



2 v^ cos^ a ' 

whence, by division, we obtain, since x^ ~ 2 x^, 

yx — Xx tana, a j - 1 n , ^V\ — Vi 

j~— — = 1 and therefore tana, a = —^ ^. 

^2 — ^2 tang, a ^ '^ x 

If in one of the foregoing formulas, instead of — ^ — , we put 1 + 

COS. Oi 

ta7ig^ a, and for tang, a we substitute the last expression, we obtain 
the required formula for the velocity of efflux 



828" GENERAL PRINCIPLES OF MECHANICS. [§410. 



_ J £^ J (1 + tang.d')gx' 

^' ~ "^ 2 (2/2 - ^ tang, a) cos^ a~ ^ ' 2 (2 ^2 - ^ 2/0 
Jg \x' + (i'y, - y,Y\ 

-^ 4(y, -2^/0 ; 

Hence the coefficient of velocity is 



^ = — 



\^¥gli 8 A (2/2 - 2 y,) ' 



Example 1) The following measurements of an uncontracted stream, 
which issued from a well-rounded orifice 1 centimeter wide, were made : 
X = 3,480 meters, 

y^^z^—z = 0,267 — 0,1135 = 0,1535 meters, 
y^=z^-z = 0,669 - 0,1135 = 0,5555 " 
and the head of water was Ti = 3,359 meters. From these data we find 
the coefficient of velocity to be 

. / 2,48^ + 0,059^ . / M 85 _o aao 

^ =^ y 8T3;359"rp485 ^ V 26;872 . 0,2485 -"'^^^• 
Since no contraction took place, a = l and therefore ju = <p. The results 
of measurements given in the remark to § 405 agree well with this value. 

3) The measurements of a perfectly contracted stream, which passed 
through a circular orifice in a thin plate, were, for a head of water h = 
3,396 meters, the following : 
X = 2,70 meters, 

y^^z^—z = 0,2465 — 0,1115 = 0,1350 meters, 
y^=z^-z = 0,6620 - 0,1115 = 0,5505 " 
whence it follows that 

_ / 2,70- + ^pl^ _ / 7;29"01 ^ 

"^ ~ r 8 . 3,396 . 0,2805 ~ ^ 37,168 . 0,2805 ' 
From the measurement of the discharge //was calculated to be = 0,617 ; 

hence the coefficient of contraction was a = - = 0,631, which agreed very 
well with the measurement of the cross-section of the stream. 

§ 410. Rectangular Lateral Orifices. — The most accurate 
experiments upon efflux through large lateral rectangular orifices 
are those made at Metz by Poncelet and Lesbros. The width of 
these apertures were 2 and in some cases 6 decimeters and their 
heights were different, varying from 1 centimeter to 2 decimeters. 
In order to produce a perfect contraction, the orifice was made in a 
brass plate 4 millemeters thick. From the results of these experi- 
ments, these savants have calculated, by interpolation, the tables, 
which are given at the end of this paragraph, and which can be 
employed for the measurement or calculation of discharges. 

If h is the width of the orifice K L, Fig. 700, and if hi and lu 



§410.] CO^ TRACTION OF THE VEIN OR JET OF WATER. 829 



are the heights E G and E L oi the level of the water above the 
lower and upper horizontal edge of the orifice, we have, according 
to § 401, the discharge 

If we introduce the height of the orifice G L = a = Jix — h<i 

7l^ + ^2 1 

, we have approxi- 



and the mean head of water EM— h = 
matively 



2 



and, therefore, the effective discharge is 

Ci = /^ = (l - of^) ^ « ^ V^gh. 
Fm. 700. If we put 

we have more simply 

Qi = Hittb V2g h, 
and as it is more convenient to employ 
this simple formula for the discharge, 
the values of /^i, and not those of /x 
are given. 

Since the water in the neighborhood 
'' of the orifice is in motion, it stands 
higher immediately in front of the 
wall, in which the aperture is made ; 
for this reason two tables are given, 
one to be used, when the heads of water are measured at a distance 
from the orifice, and the other, when they are measured directly at 
the wall of orifice. We see from both these tables that, with some 
exceptions, the less the height of the orifice and the head of water 
is, the greater the coefficient of efflux is. 

If the width of an orifice is different from those given, we 
must employ these coefficients to calculate the discharge, as we 
have no other experiments to base our calculations upon. That 
we are not liable to great error can be seen by comparing the co- 
efficients for the orifices, whose widths are 0,6 meters, with those, 
whose widths are 0,2 meters, for the same head of water. If 
the apertures are not rectangular, we determine their mean heighl 
and width and substitute in the calculation the coefficient corre- 
sponding to these dimensions. It is always better to measure the 
head of water at a great distance from the orifice and to employ 




830 GENERAL PRINCIPLES OF MECHANICS. [§410. 

the first table than to measure it immediately at the orifice, where 
the surface of the water is curved and less tranquil than at a dis- 
tance from it. 

Example — 1) What is the discharge through an orifice 2 decimeters 

wide and 1 decimeter high, when the surface of the water is 1^ meters 

above the upper edge ? Here we have 

h, + A„ 1,6 + 1,5 
d = 0,2, a = 0,1, A = ' ^ = ' ^ ' = 1,55 meters, 

and, therefore, the theoretical discharge is 

^ = 0,1 . 0,2 V2^ Vl,^ = 0,02 . 4,429 . 1,245 = 0,1103 cubic meters. 
But Table I gives for a = 0,1 and Ag = 1,5, ju^ = 0,611, hence the 
real discharge is 

Q = 0,611 . 0,1103 = 0,0674 cubic meters. 

2) What is the discharge through a rectangular orifice in a thin plate, 
whose height is 8 inches and whose width 2 inches, under a head of water 
of 15 inches above the upper edge ? The theoretical discharge is 

Q = I . 1 . 8,025 Vf = 0,8917 . 1,1547 = 1,0296 cubic feet. 
But two inches is about 0,05 meters and 15 inches about 0,4 meters, 
we can therefore take the value /u^ = 0,628, corresponding to a = 0,05 and 
h^ = 0,4, and put the required discharge 

Q^ = 0,628 . 1,0296 = 0,647 cubic feet. 

3) If the width is 0,25 meter, the height 0,15 and the head of water 
^3 = 0,045, we have 

Q = 0,25 . 0,15 . 4,429 VO^S = 0,166 . 0,3464 = 0,0575 cubic meters ; 
the height 0,15 corresponds, for 7i^ = 0,04, to the mean value 

.,=5^^^^^ = 0,5935, 
and, for h^ = 0,05, to 

.,=5^-°^ = 0,595. 

Now since h^ = 0,045 is given, we substitute the new mean 

0,5925 + 0,5950 ^ ^^^ 
-g = 0,594 

as coefficient of efflux, and we obtain the required discharge 
Q^ = 0,594 . 0,0575 = 0,08415 cubic meters. 
Remark. — The coefficients of velocity do not change sensibly for a rec- 
tangular orifice, when we change the height into the width or vice versa, 
as is demonstrated by the following experiments of Lesbros (see his '' Ex- 
periences Hydrauliques, Paris, 1851"). 

An orifice 0,60 meters wide and 0,02 meters high, under a head of water 
from li = 0,30 to 1,50 meters, gave 

fi^= ix=z 0,635 to 0,622, 
and, on the contrary, when it was set on edge, or when the height was 0,60 
meters and the width 0,02 meters, 

^1 = 0,610 to 0,626 and 
ft = 0,638 to 0,627. 



410.] CONTRACTION OF THE VEIN OR JET OF WATER. 



831 



TABLE I. 

The coefficients of efflux of water issuing from rectangular orifices in a thin 
mrtical plate^ according to Poncelet and Lesbros. 
(The heads of water are measured above the orifice at a point where the 
water can be considered as still. The values below the asterisk (*) are de- 
termined only by interpolation.) 



Head of water or dis- 
tance of the level of 
the water above the 
upper edge of the 
orifice, in meters. 






HEIGHT 


OF THE ORIFICE, IN 1 


WETERS. 






Width of the orifice = 0,2 meters. 


Width of the orifice 
= o,6 meters. 


0,20 


0,10 


0,05 


0,08 


0,02 


0,01 


0,20 


0,02 


0,000 


u 


a 


a 


a 


a 


i( 


u 


u 


0,005 


u 


a 


(( 


u 


u 


0,705 


u 


(( 


0,010 


u 


u 


0,607 


0,680 


0,660 


0,701 


a 


0,644 


0,015 


u 


0,593 


0,612 


0,632 


0,660 


0,6S7 


((. 


0,644 


0,020 


0,572 


0,596 


0,615 


0,684 


0,659 


0,694 


a 


0,643 


0,030 


0,578 


0,600 


0,620 


0,688 


0,659 


0,688 


0,593 


0,642 


0,040 


0,582 


0,603 


0,628 


0,640 


0,658 


0,683 


0,595 


0,642 


0,050 


0,585 


0,605 


0,625 


0,640 


0,658 


0,679 


0,5G'7 


0,641 


0,060 


0,587 


0,607 


0,627 


0,640 


0,657 


0,676 


0,569 


0,641 


0,070 


0,588 


0,609 


0,628 


0,639 


0,656 


0.678 


0,600 


0.640 


0,080 


0,589 


0,610 


0,629 


0,638 


0,656 


0,670 


0,601 


0^640 


0,090 


0,591 


0,610 


0,629 


0,637 


0,655 


0,668 


0,601 


0,639 


0,100 


0,592 


0,611 


0,630 


0,637 


0,654 


0,666 


0,GC2 


0.639 


0,120 


0,593 


0,612 


0,630 


0,636 


0,658 


0,668 


0,603 


0,638 


0,140 


0,595 


0,618 


0,630 


0,685 


0,651 


0.660 


0,603 


0,637 


0,160 


0,596 


0,614 


0,681 


0,684 


0,650 


0,658 


0,604 


0,637 


0,180 


0,597 


0,615 


0,630 


0,684 


0,649 


0,657 


0,605 


0,636 


0,200 


0,598 


0,615 


0,630 


0,638 


0,648 


0,655 


0,605 


0,685 ' 


0,250 


0,599 


0,616 


0,630 


0,682 


0,646 


0,653 


0,606 


0,634 


0,300 


0,600 


0,616 


0,629 


0,682 


0,644 


0,650 


0,607 


0,683 


0,400 


0,602 


0,617 


0,628 


0,631 


0,642 


0,647 


0,607 


0,631 


0,500 


0,603 


0,617 


0,628 


0,680 


0,640 


0,644 


C.607 


0,630 


0,600 


0,604 


0,617 


0,627 


0,680 


0,638 


0,642 


0,607 


0,629 


0,700 


0,604 


0,616 


0,627 


0,629 


0,637 


0,640 


0,607 


0,628 


0,800 


0,605 


0,616 


0,627 


0,629 


0,686 


0,637 


0,606 


0,628 


0,900 


0,605 


0.615 


0,626 


0,628 


0,684 


0,635 


0,606 


0,627 


1,000 


0.605 


0,615 


0,626 


0,628 


0,683 


0,632 


0.605 


0,626 


1,100 


0^604 


0,614 


0,625 


0.627 


0,631 


0,629 


0,604 


0,626 


1,200 


0,604 


0,614 


0,624 


0,626 


0,628 


0,626 


0,604 


0,625 i 


1,300 


0,603 


0,613 


0,622 


0,624 


0,625 


0,622 


0,603 


0,624 : 


1,400 


0,603 


0,612 


0,621 


0,622 


0,622 


0,618 


0,603 


0,624 


1,500 


0,602 


0,611 


0,620 


0,620* 


0,619* 


0,615* 


0,602 


0,623 


1,600 


0,602 


0,611 


0,618 


0,618 


0,617 


0,618 


0,602* 


0.623 


1,700 


0,602* 


0,610* 


0,617 


0,616 


0,615 


0,612 


0,602 


0,622 


1,800 


0,601 


0,609 


0,615* 


0,615 


0,614 


0,612 


0,602 


0,621*; 


1,900 


0,601 


0,608 


0,614 


0,618 


0,612 


0,611 


0,602 


0,621 


2,000 


0,601 


0,607 


0,613 


0,612 


0,612 


0,611 


0,602 


0,620 


3,000 


0,601 


0,603 


0,606 


0,608 


0,610 


0,609 1 


0,601 


0,615 



Similar tables for the 
the Ingenieur, page 432. 



Prussian system of measures are to be found in 



832 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 410, 



TABLE XL 

TJie coefficients of efflux of water issuing from rectangular orifices in a tliin 
vertical plate^ a/xording to Poncelet and Lesbros. 

(The heads of water were measured directly at the orifice. The values 
above and below the asterisks (*) are determined by interpolation only.) 



;er or dis- 
le surface 
er above 
:dge of the 
neters. 


HEIGHT OF THE ORIFICE, IN METERS. 












Width of the 


5"" 5 ^^ r. 




Width of the orifice = 0,2 meters. 




orifice 


&^&^.s 












= 0,6 meters. 


d of 
nee < 
the 
eup; 
ifice, 






























0,20 


0,10 


0,05 


0,03 


0,02 


0,01 


0,20 


0,000 


0,619 


0,667 


0,713 


0,776 


0,783 


0,795 


0,586 


0,005 


0,597 


0,630* 


0,668* 


0,725* 


0,750* 


0,778* 


0,587 


0,010 


0,595 


0,618 


0,642 


0,687 


0,720 


0,762 


0,589 


0,015 


0,594 


0,615 


0,639 


0,674 


0,707 


0,745 


0,590 


0,020 


0,594* 


0,614 


0,638 


0,668 


0,697 


0.729 


0,591 


0,030 


0,593 


0,613 


0,637 


0,659 


0,685 


0,708 


0,592 


0,040 


0,593 


0,612 


0,636 


0,654 


0,678 


0,695 


0,594* 


0,050 


0,593 


0,612 


0,636 


0,651 


0,672 


0,686 


0,595 


0,060 


0,594 


0,613 


0,635 


0,647 


0,668 


0,681 


0,596 


0,070 


0,594 


0,613 


0,635 


0,645 


0,665 


0,677 


0,597 


0,080 


0,594 


0,613 


0,635 


0,643 


0,662 


0,675 


0,598 


0,090 


0,595 


0,614 


0,634 


0,641 


0,659 


0,672 


0,599 


0,100 


0,595 


0,614 


0,634 


0,640 


0,657 


0,669 


0,600 


0,120 


0,596 


0,614 


0,633 


0,637 


0,655 


0,665 


0,601 


0,140 


0,597 


0,614 


0,632 


0,636 


0,653 


0,661 


0,602 


0,160 


0,597 


0,615 


0,631 


0,635 


0,651 


0,659 


0,602 


0,180 


0,598 


0,615 


0,631 


0,634 


0,650 


0,657 


- 0,603 


0,200 


0,599 


0,615 


0,630 


0,833 


0,649 


0,656 


0,603 


0,250 


0,600 


0,616 


0,630 


0,632 


0,646 


0,653 


0,604 


0,300 


0,601 


0,616 


0,629 


0,632 


0,644 


0,651 


0,605 


0,400 


0,602 


0,617 


0,629 


0,631 


0,642 


0,647 ■ 


0,606 


0,500 


0,603 


0,617 


0,628 


0.630 


0,640 


0,645 


0,607 


0,600 


0,604 


0,617 


0,627 


0,630 


0,638 


0,643 


0,607 


0,700 


0,604 


0,616 


0,627 


0,629 


0,637 


0,640 


0,607 


0,800 


0,605 


0,616 


0,627 


0,629 


0,636 


0,637 


0,607 


0,900 


0,605 


6,615 


0,626 


0,628 


0,634 


0,635 


0,607 


1,000 


0,605 


0,615 


0,626 


0,628 


0,633 


0,632 


0,606 


1,100 


0,604 


0,614 


0,625 


0,627 


0,631 


0,629 


0,606 


1,200 


0,604 


0,614 


0,624 


0,626 


0,628 


0,626 


0,605 


1,300 


0,603 


0,613 


0,622 


0,624 


0,625 


0,622 


0,604 


1,400 


0,603 


0,612 


0,621 


0,622 


0,622 


0,618 


0,603 


1,500 


0,602 


0,611 


0,620 


0,620* 


0,619* 


0,615* 


0,603 


1,600 


0,602 


0,611 


0,618 


0,618 


0,617 


0,613 


0,602 


1,700 


0,602* 


0,610* 


0,617 


0,616 


0,615 


0,612 


0,602 


: 1,800 


0,601 


0,609 


0,615* 


0,615 


0,614 


0,612 


0,602 


i 1,900 


0,601 


0,608 


0,614 


0,613 


0,613 


0,611 


0,602 


2,000 


0,601 


0,607 


0,614 


0,612 


0,612 


0,611 


0,602 


1 3,000 


0,601 


0,603 


0,606 


0,608 


0,610 


0,609 


0,601 



§411.] CONTRACTION OF THE VEIN OR JET OF WATER. 



833 




§ 411. Overfalls. — If the water flows through an overfall, weir 
ornotch (Fr. deversoirs; Ger. Ueherfalle) in a thin wall, as, e.g., F B, 

Fig. 701, the stream is contracted 
Fig. 701. on three sides and a diminution 

of the discharge is produced. The 
discharge through this orifice is 

Here the head of water E H = Ji 
is to be measured, not at the edge, 
but at least three feet from the 
wall in which the notch is cut ; for the surface of the. water is de- 
pressed immediately behind the orifice, and the depression increases 
continually towards the orifice, and in the plane of the orifice its 
yalue G R is from 0,1 to 0,25 of the head of water F R, so that the 
thickness F G of the stream is but 0,9 to 0,75 of the head of water. 
Many experiments have been made upon efilux of water through 
notches in a thin j^lctte, and the results, although very multifarious, 
do not agree as well as could be desired. The following tables con- 
tain the results of the experiments of Poncelet and Lesbros. 

1. TABLE OF COEFFICIENTS OF EFFLUX FOR OVERFALLS 
TWO DECIMETERS WIDE, ACCORDING TO PONCELET AND 
LESBROS. 



Head of water A 
in meters. 


0,01 


0,02 


0,03 


0,04 


0,06 


0,08 


0,10 


0,15 


0,20 


0,22 


Coefficient 
of efflux 


0,424 


0,417 


0,412 


0,407 


0,401 


0,397 


0,395 


0,393 


0,390 


0,385 



2. TABLE OF THE COEFFICIENTS OF EFFLUX FOR OVERFALLS 

SIX DECIMETERS WIDE. 



Head of water A 
in meters. 


0,06 


0,08 


0,10 


0,12 


0,15 


0,20 
0,395 


0,30 


0,40 


0,50 
0,391 


0,60 


Coefficient 
1 of efflux 

1 


0,412 


0,409 


0,406 


0,403 


0,400 


0,391 


0,391 


0,390 



Hence for approximate determinations we can put f-i^ = 0,4. 
53 



834 GENEIIAL PRINCIPLES OF MECHANICS. [§ 413. 

Eytehvein found, by Ms experiments with oyerfalls of great width,, 
the mean value of i^i tohe = j [jl — 0,42, and Bidone fii = | 0,62 
— 0,41, etc. The most extensive experiments were made by d'Au- 
buisson and Castel. Erom these d'Aubuisson concludes that for 
overfalls, whose width is not greater than -3 that of the canal or of 
the wall in which the weir is placed, we can put fi = 0,60 or ^ fi = 
0,40 ; that, on the contrary, when the overfall extends across the 
whole wall or has the same width as the canal, we must take ft =: 
0,665 or ftj =: 0,444 ; that, finally, when the relations between the 
width of the notch and that of the canal differ from the above, the 
coefficient of efflux is very varied, the extremes being 0,58 and 0,66. 
The experiments made in 1853 and 1854, at Hanswyk, upon over- 
falls 3 to 6 meters wide under a head of 0,1 to 1,0 meters gave 
II =--:: 0,64 to 0,65 or I ft = 0,427 to 0,433 (see the " Zeitschrift des 
Archit- und Ingen-Yereins fur Hanover, 1857"). The researches 
made by the author upon the efflux of water through overfalls re- 
fer the variation of these coefficients of efflux to certain laws, which 
will be noticed further on (§ 417). 

Example — 1) The discharge per second of an overfall, 0,25 meters 
wide under a head of water of 0,15 meters is 



Q = 0393 .dh^/2g h = 0,393 . 4,429 . 0,25 (0,15)1 = 0,435 . 0,0581 
= 0,02527 cubic meters. 

2) What must be the width of an overfall, which under a head of water 
of 8 inches will discharge 6 cubic feet of water ? Here we have 

7, — ^ ^ _ ? Q 434 feet 

/"l^/27I^ 0,4 . 8,025 V(|f 3,210.0,5443 

If according to Eytelwein we take /^i = 0,42, we have 

6 
^ == 3;37To;5443 = ^'^'^^• 

§ 412. Maximum and Minimum Contraction. — When wa- 
ter flows through an orifice in a plane surface, the axis of the ori- 
fice is at right angles to the wall of the vessel and we have a me- 
dium contraction ; if, however, the axis of the orifice or of the 
stream forms an acute angle with the portion of the wall of the 
vessel containing the aperture, the contraction is smaller, and if 
the angle between this axis and the inner surface of the vessel is 
obtuse, the contraction is greater. The first case is represented in 
Fig. 702 and the second in Fig. 703. This difference of contrac- 
tion is, of course, due to the fact that in the former case the 
molecules of the water, which are flowing towards the orifices, are 



^ 413.] CONTRACTION OF THE VEIN OR JEf OF WATER. 335 



deviated less, and in the latter case more, from their primitive di- 
rection, while passing through this aperture and forming the vein. 
The contraction is a minimum, i.e., null, if, by gradually con- 
tracting the wall surrounding the orifice, the water is prevented 
from flowing in upon the side and, on the contrary, a maximum 
when the direction of the wall is opposite to that of the stream, so 
that certain molecules must describe an angle of 180 degrees in 



Fig. 703. 



Fig. 703. 



Fig. 704. 



Fig. 705. 







order to reach the orifice. Both cases are represented in Figures 
704 and 705. In the first case the coefficient of efflux is nearly 1, 
viz. : 0,96 to 0,98, and in the second case, according to the measure- 
ments of Borda, Bidone and of the author, its mean value is = 0,53. 
In iDractice, variations of the coefficients of efflux, produced by 
convergent walls, often occur, particularly in the case of sluices, 
which are inclined to the horizon, as is shown in Fig. 706. Pon- 
celet found for such an orifice the coefficient of efflux [i ■=. 0,80, 
when the gate was inclined at an angle of 45°, and, on the contrary, 
ju is only — 0,74, when the inclination is 63A degrees, i.e., for a 
Fig. 706. Fig. 707. 





D C 

slope of one-half to one. For the overfall, represented in Fig. 707, 
where, as in Poncelet's sluice, contraction takes place upon one 
side only, the author found \i — 0,70 or \i^ = :-; \l — 0,467 for an 
inclination of 45°, and \i = 0,67 or fi^ = 0,447 for an inclination 
of 683 degrees. 

According to M. Boileau (see his Traite de la mesure des eaux 



83i» 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 413. 



courantes) we can put for an oyerfall, which is inclined upwards 
in such a way that the horizontal projection is ^ the vertical, or 
that the angle of inclination is 71 J degrees, the coefl&cient of efflux 
= 0,973 times the coefficient of efflux for an overfall with a vertical 
wall. We also find from the experiments of Boileau that, for ver- 
tical overfalls placed at an angle to the direction of the stream, we 
must put, when the angle is 45°, the coefficient of efflux = 0,942 
and, when the angle is 65°, only 0,911 times the coefficient of efflux 
for the normal overfall ; the whole length of the edge, over which 
the water flows, being of course considered as the length of the 
orifice. 

Example. — If a sluice gate, which is incfined at an angle of 50 degrees 
and closes a trough 2i feet wide, is raised ^ foot and if the surface of the 
water then stands j)ermanently 4 feet above the bottom of the trough, the 

height of the orifice is 

a = ^ sin. 50° = 0,3830, 
the head of water is 

A = 4 - ^ , 0,3830 = 3,8085 feet, 
and the coefficient of velocity is // = 0,78, hence the discharge is 

Q = 0,78 . 2,25 . 0,3830 . 8,025 V3,8085 = 10,52 cubic feet. 

§ 413. Scale of Contraction. — The more the direction of the 
water ivhich fiotus in from the sides differs from that of the stream, 
the greater is the contraction of the vein. 

When a stream flows through the orifice C, Fig. 708, in a plane 
thin plate, the angle (5, formed by its axis or direction of motion 

Fig. 708. 




with that of the molecules of water which flow in from the side, is 

a right angle f ^ ) J when the orifice A is formed by the thin edge 

of a tube, this angle 6 is two right angles ( tt ) ; when we have 
a conical divergent mouth-piece B, 6 is between A tt and tt- 
when the discharge takes place through a conical convergent 



§414] CONTRACTION OF THE VEIN OR JET OF WATER. 837 



77 



mouth-piece, 6 is between and - ; and when a. cylindrical mouth- 

piece B well rounded off internally is used, it is = 0. 

In order to discover the law, according to which the contraction 
diminishes with the angle 6, the author made a series of experi- 
ments with a great number of mouth -pieces 2 centimeters wide and 
under different pressures (from 1 to 10 feet) ; the results of these 
experiments are given in the following table : 



6 


f ■ — 

180° 157^» 


1350 


112|.o 


900 


67|o 


450 


2210 


11^ 


5fo 


00 


fJ- 


0,541 


0,546 


0,577 


0,603 0,632 0,684 0,753 0,882 


0,924 


0,9490,966 



This table gives, it is true, only the coefficients of efflux i-i corre- 
sponding to different angles of deviation (5; the coefficients of 
contraction are from 1 to 2 per cent, greater, since a small loss of 
velocity always takes place during the efflux. In order to prevent 
any loss of vis viva, when the water enters the mouth-pieces D and 
E, the latter are rounded off at the entranf^e. The friction, to be 
overcome by the water in passing along the walls of the mouth- 
piece, will be determined in the following chapter. 

Remark. — According to the calculations of Prof. Zeuner (see Civilin- 

genieur, Vol. 2d, page 55) of the results of the above experiments, we can 

put 

^g = /z, ^ (1 + 0,33214 {COS. 6y + 0,16672 {cos. 6)') 

/.^x ^ denoting the coefficient of efflux for an orifice in a plane thin plate, 

for which the maximum deviation of the elements of the water during efflux 

is = |- 7r = 90°, and z^^, on the contrary, denoting the coefficient of efflux for 

an orifice in a conical thin plate, where the maximum deviation of the 

elements of the water upon entering is S. 

§ 414. Partial or Incomplete Contraction. — We have as 
yet studied only the case, where the water flows in from all sides of 
the opening and forms a stream contracted upon all sides ; we must 
now consider the case, where the water flows in from but one or 
more sides to the orifice, and consequently produces a stream which 
is incompletely contracted. In order to distinguish these condi-' 
tions of contraction from each other, we will call the case, where 
the stream is contracted on all sides, complete contraction, and the 
case, where the stream is contracted upon a part only of . its 
periphery, partial or incomjMe contraction (Fr. contraction incom- 
plete ; Ger. unvollstandige or partielle Contraction). Incomplete 
contraction occurs whenever an orifice in a tliin plane plate is 



838 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 414. 



surrounded upon one or more sides by a plate placed in the 
direction of the stream. In Fig. 709 there are represented four 
orifices a, h, c, d of equal magnitude in the bottom ^ (7 of a vessel. 
The contraction of the water flowing through the orifice a in the 
middle of the bottom is complete, for in this case the water can 
flow in from all sides ; the contraction of the stream in passing 
through h, c ox d is incomplete, for the water in these cases can 
flow in from only three, two or one side. In like manner, 
when a rectangular lateral orifice extends to the bottom of the 
vessel, the contraction is incomplete ; for that upon the side of the 
base is wanting ; if further the opening extend to the bottom and 
sides of the trough, there will be contraction upon one side only. 

Incomplete contraction manifests itself in two ways. First, it 
gives an inclined direction to the stream ; and secondly, it causes a 
greater discharge. 

Fig. 709. Fig. 710. 

B 



.■>^,.SS^S^.V,N^^,S^VV 



t J 



I! 



■sw^.^vv.. ^v^^^.v^„■s,v«^^ss■N^s^^ss^N,^.^^^^N^^^^^^^V^■ 




D 



If, E.G., the lateral orifice F, Fig. 710, reaches to the bottom 
G D, so that no contraction can take place there, the axis F G oi 
the stream will form an angle H F G of about 9 degrees with the 
normal F H to the plane of the orifices. This deviation of the 
stream becomes much greater when two adjoining sides are con- 
fined. If the orifice has a border upon two opposite sides, the con- 
traction at those points is thereby prevented, and this deviation of 
the stream does not take place, but at a certain distance from the 
orifice the stream becomes wider than it would have done, if it had 
not been confined upon those sides. Although a greater discharge 
is obtained when the contraction is incomplete, yet it is generally to 
be avoided, since it is always accompanied by a deviation in the 
direction and by a great increase in the width of the stream. 
Experiments -upon the eflaux of water, when the contraction is 
incomplete, have been made by Bidone and by the author. 



§ 414] CONTRACTION OF THE VEIN OR JET OF WATER. 839 

Their results show that the coefficient of efflux increases very 
nearly with the ratio of the length of the border to the entire peri- 
phery of the orifice ; but it is easy to perceive that this relation is 
different, when the periphery is nearly or entirely surrounded by a 
border, in which case the contraction is almost or totally done away 
with. K we put the ratio of the portion with a rim to the entire 
periphery = n and denote by a an empirical quantity, we can put, 
approximatively, the ratio of the coefficient //„ of efflux for incom- 
plete contraction to the coefficient \i^ for complete contraction 

-^ = 1 + «: ^, and consequently \h^ — {\. -\- a n) \jl^. 

Bidone^s experiments gave for small circular orifices k = 0,128, 
and for square ones fc = 0,152 ; those of the author gave for small 
rectangular orifices a:= 0.134, and for larger ones (Poncelet's mouth- 
pieces) 0,2 meter wide and 0,1 meter high fc = 0,157 (see the Maga- 
zine "der Ingenieur," vol. 2d). In practice rectangular orifices 
with rims are almost the only ones employed ; we will assume for 
them, as a mean value, k = 0,155, and consequently put 

/x„ = (1 + 0,155 n) II,. 

For a rectangular lateral orifice, whose height is a and whose 

width is ^, we have n = ^n tt? when there is no contraction 

^ {a + by 

upon the side b, if, e.g., this side is upon the bottom ; 7^ = ^, when 

one side a and one side b are provided with rims ; and n = ^r—, ^r, 

^ ' 2 {a + b) 

when the contraction is prevented upon the side b and upon the 

two sides a, the latter case occurs, when the orifice occupies the 

entire width of the reservoir and extends to the bottom. 

Example. — What is the discharge through a vertical sluice 3 feet wide 
and 10 inches high, when the head of water is 1^ feet above the upper 
edge of the orifice and the lower edge is at the bottom of the trough, so 
that there is no contraction upon that side ? The theoretical discharge is 

Q = ^.d . 8,025 VI, 5 + tV = f • 8,025 Vl,9166 = 27,77 cubic feet. 
According to Poncelet's table for perfect contraction /u. = 0,'!504, but 

we have 

8 9 



n = 



— 9 



3 (3 + i|) 18 + 5 ^' 
hence for the present case of incomplete contraction 

//„=:(! + 0,155 . ^) . 0,604 = 1,060 . 0,604 = 0,640 

and the efiective discharge is 

Q = 0,640 Q = 0,640 . 27,77 = 17,77 cubic feet 



840 GENERAL PRINCIPLES OF MECHANICS. [§415. 

§ 415. Imperfect Contraction. — The contraction of the vein 
depends also upon this fact, viz, : whether the water is sensibly at 
rest in front of the orifice or whether it arrives there with a certain 
velocity ; the faster the water approaches the orifice of efflux, the 
less the stream is contracted, and consequently the greater is the 
discharge. The various relations of contraction and efflux, given 
and discussed m what precedes, are applicable only where the ori- 
fice is m a large wall, in which case we can assume that the water 
arrives at the orifice with a very small velocity , we must now 
investigate the relations of contraction and efflux, when the cross- 
section of the orifice is not much smaller than that of the approach- 
ing water, in which case the water arrives with a velocity, which is 
not negligable. In order to distinguish these two cases from each 
other, let us call the contraction which occurs, when the water 
above the orifice is at rest, perfect contraction and that which 
occurs, when the water is in motion, imperfect contraction (Fr con- 
traction imparfaite ; Ger. unvollkommene Contraction). The 
contraction during efflux from the vessel A C, Fig, 711, is imper- 
fect ; for the cross-section F of the orifice is not 
Fig. 711. much smaller than that G of the water approach- 

ing it or the area of the wall C D, in which tins 
orifice is placed. If the vessel was of the form 
A B Ci Di and the area of the base Ci D^ was 
much greater than that of the orifice F, the 
efflux would take place with perfect contraction. 
The imperfectly contracted stream is distm- 
Dr'D^ife^e-C^ guished from the perfectly contracted one not 
Eiil only by its size, but also by the fact that it is not 

so transparent and crystalline as the latter is. 

If we denote the ratio of the area F of the orifice to that G of 

F 
the wall in which it is situated, or -^, by n, the coefficient of efflux for 

perfect contraction by ii^ and that for imperfect contraction by m^, 
we can put with great accuracy, according to the experiments and 
calculations of the author, 

1) for circular orifices 

^n =^ ^0 [1- + 0,04564 (14,821" - 1)], 

2) and for rectangular orifices 

/z„ = ^0 [1 + 0,0760 (9-^ - 1)].* 

* Experiments upon the imperfect contraction of water, etc., Leipzig, 1843. 




§415.] CONTRACTION OF THE VEIN OR JET OF WATER. 841 

In order to facilitate the calculations which are required in 

practice, the corrections — of the coefficient of efflux in con- 

sequence of the imperfect contraction have been arranged in the 
following tables : 

TABLE I. 

The corrections of the coefficients of efflux for circular orifices. 



n 


0,05 


0,10 


0,15 
0,023 


0,20 


0,25 


0,30 


0,35 


0,40 


0,45 


0,50 


f^n — ^0 


0,007 


0,014 


0,034 


0,045 


0,059 


0,075 


0,092 


0,112 


0,134 


n 


0,55 


0,60 


0,65 


0,70 


0,75 


0,80 


0,85 


0,90 


0,95 


1,00 


/^ — ^0 


0,161 


0,189 


0,223 


0,260 


0,303 


0,351 


0,408 


0,471 


0,546 


0,631 



TABLE II. 

The corrections of the coefficients of efftuxfor rectangular orifices. 



n 


0,05 


0,10 


0,15 


0,20 


0,25 


0,30 


0,35 


0,40 

0,107 


0,45 
0,128 


0,50 


U-n — Up 


0,009 


0,019 


0,030 


0,042 


0,056 


0,071 


0,088 


0,152 


n 


0,55 


0,60 


0,65 


0,70 


0,75 


0,80 


0,85 


0,90 


0,95 


1,00 


Hn — Ho 
^0 


0,178 


0,208 


0,241 


0,278 


0,319 


0,365 


0,416 


0,473 


0,537 


0,608 



The upper lines in these tables contain yarious values of the 

F 
ratio YT of ^^6 cross-sections, and immediately below are the corre- 
sponding additions to be made to the coefficient of efflux on account 
of the imperfect contraction, e.g., for the ratio 7i = 0,35, i.e., for 
the case, where the area of the orifice is 35 hundredths of the area 
of the entire wall, in which the orifice is made, we have for a cir- 
cular orifice fi^ — ji^ 



H'c 



0,075, 



and for a rectangular one = 0,088 ; the coefficient of efflux for 



842 



GENERAL PRINCIPLES OF MECHANICS 



[§ 416. 



perfect contraction must be increased in the first case 75 thou- 
sandths and in the second 88 thousandths, when we wish to obtain 
the corresponding coefficient of efflux for imperfect contraction. 
If the coefficient of efflux were = 0^615, we would have in the 

first case 

Mo35z= 1,075 . 0,615 = 0,661 

and in the second case 

it^o,35 = 1,088 . 0,615 = 0,669. 

Example. — What is the discharge through a rectangular lateral orifice 
F^ which is li feet wide and | foot high, when it is cut ia a rectangular 
wall C Z), Fig. 713, 2 feet wide and 1 foot high, and when the head of 

water E H — Ti, where the water is at 
rest, is 2 feet The theoretical dis- 
charge is 
Q = 1,25 . 0,5 . 8-035 V2 

= 5,0156 , 1,414 = 7,093 cubic feet, 
and the coefficient of efflux for perfect 
contraction is, according to Poncelet, 

K, = 0,610, 
but the ratio of the cross-sections is 
F 1,35 . 0,5 

and for n = 0,312 we have, according to Table II, page 841 

^" ~ """ = 0,071 + il (0,088 - 0,071) = 0.071 + 0,004 = 0,075; 

hence the coefficient of effl-ux for the present case is 

/"o,3i2 = 1.075 . 11, = 1,075 + 0,610 = 0,6557, 

and the effective discharge is 

Q^ = 0,6557 . Q = 0,6557 . 7,092 = 4,65 cubic feet. 

§ 416. Efflux of Moving "Water. — We have heretofore 
assumed that the head of ivater was measured in still loater ; we 
must now discuss the case where the head of water can be meas- 
ured only m water, which is approaching the orifice with a certain 
velocity. If we assume the orifice to be rectangular and denote 
the width by b, the head of water in reference to the two horizon- 
tal edges by /?, and h^ and the height due to the velocity of ap- 
proach c of the water by k, we have the theoretical discharge 

Q = 11^^ {{Ih + hf - (h, + k)^. 
This formula cannot be directly employed for the determination 
of the discharge, since the height due to the velocity 




^9 






'^9 



416.] CONTRACTION OF THE VEIN OR JET OF WATER. 843 



depends also upon Q, and, if we transform it, we obtain a compli- 
cated equation of a high degree ; it is much simpler, therefore, to 
put the effective discharge 

Qi = [i,a i V2gh 
and to understand by ju^i not a simple coefficient of efflux, but a co- 
efficient depending principally upon the ratio of the cross-sections. 
This case is often met with in practice, e.g., when we wish to 
measure the quantity of water which passes through a ditch or 
canal ; for we can seldom dam up the water by means of a trans- 
verse wall B (7, Fig. 713, to such a height that the area F of the 

orifice, through which the water 
is discharged, will be but a small 
fraction of the cross-section of the 
stream which approaches it, and 
it IS only m the latter case that 
the velocity of approach is very 



small compared to the mean ve- 
locity of efflux. 
In the experiments made by the author with Poncelet's orifices 

the head of water was measured 1 meter back from the plane of the 

orifice , they gave 



Fig. 713. 




F 



^0 



= 0,641 (^ 



(o) = «'^^i 



n% 



n = -^ denoting the ratio of the cross-sections, which should not 

be much greater than 4, jli^ denoting the coefficient of efflux for 
perfect contraction, taken from Poncelet's table, and jifc„ the coeffi- 
cient of efflux for the present case. Let b be the width and a the 
height of the orifice, di the width and «, the depth of the stream 
of water and h the depth of the upper edge of the orifice below the 
level of the water, then we have the effective discharge 



e, = ft . « 5 [i + 0,641 ( Jl)'] /a SI (a + I). 

The following table is useful in abridging calculations in practice. 



n 


0,05 
0,002 


0,10 
0,006 


0,15 
0,014 


0,20 
0,026 


0,25 
0,040 


0,30 



0,058 


0,35 
0,079 


0,40 
0,103 


0,45 
0,130 


0,50 
0,160 


^0 



Example. — In order to find the amount of water brought by a ditch 3 
feet wide, a transverse wall, containing a rectangular orifice 2 feet wide and 



844 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 417. 



1 foot high is put in it, and the water is thus raised so that, when its level 
becomes constant, it is at a distance of 21 above the bottom and If feet 
above the lower edge of the orifice. The corresponding theoretical dis- 
charge is 

Q = a'b \l¥gli = 1,2. 8,025 VT,25 = 16,05 . 1,118 = 17,94 cubic feet. 
As the coefficient of efflux for perfect contraction is 0,602 and the ratio 
of the cross-sections is 

F ah 1.2 ^ ^^^ 

^ = ¥ = ^7&; = 2-;25:3 = ^'^^^' 

we have the coefficient of efflux in the present case 

//^ = (1 + 0,641 . 0,296^^) /z„ = 1,056 . 0,602 = 0,6357, 
and the effective discharge • 

§1 = 17,94 . 0,6357 = 11,4 cubic feet. 

§417. The contraction is also imperfect when water is dis- 
charged through overfalls (like that m Fig. 714), if the cross- 
jij.^ Yj^ section i^of the stream pass- 

ing over the sill C' is a notable 
fraction of the cross-section G 
of the approaching water. The 
overfall may extend over but 
a portion or oyer the whole of 
the canal or ditch. In the 
latter case, as there is no contraction upon the sides of the orifice, 
the discharge is greater than through orifices of the first kind. 
The author has made experiments upon these cases of efflux and 
deduced from the results obtained formulas, by means of which the 
coefficient of efflux can be calculated with sufficient accuracy, when 

F 
the ratio ^z =: — of the cross-sections is known. 

Cr 

Let li be the head of water E H above the sill of the overfall, 
ttx the total depth of water, h the width of the overfall, and hi that 
of the approaching water ; we have then 

F hh ^ 
w = 77 = — 7", and 

Or «i Ox 




\i ^\\\\\\\ \\\\\\\\\\\\\\\\sfff>.\\\\^^^^ 



1) for Poncelefs overfall 






= 1,718 



(§)■= ■• 



718 7i%- 



on the contrary, 

2) for an overfall occupying the whole width of the ditch or trough 

^n — ^0 



^0 



0,041 -h 0,3693 rf ; 



§ 417.] CONTRACTION OF THE VEIN OR JET OF WATER. 845 
hence the discharge in the first case is 

«nd in the second case, 

Q, = lfi,.b [l,041 + 0,3693 ( ^) ] ^27^, 

h denoting the head of water U H above the sill F of the overfall, 
measured at a point about one meter back of it. 

In the following tables the corrections — — — for the simplest 

values of n are given. 

TABLE I. 

Corrections of the coefficients of efflux for Poncelefs overfalls. 



n 


0,05 


0,10 
0,000 


0,15 


0,20 


0,25 


0,30 


0,35 


0,40 


0,45 


0,50 


f^n — /^„ 
^0 


0,000 


0,001 


0,003 


0,007 


0,014 


0,026 


0,044 


0,070 


0,107 



TABLE n. 

Corrections for overfalls extending over the entire widt\ or without lateral 

contraction. 



n 


0,00 


0,05 
0,042 


0,10 


0,15 


0,20 


0,25 


0,30 


0,35 


0,40 


0,45 


0,50 


t^ — fJ-o 


0,041 


0,045 


0,049 


0,056 


0,064 


0,074 


0,086 


0,100 


0,116 


0,133 



Example. — In order to determine the amount of water carried by a 
canal 5 feet wide, we place in it a transverse partition with the upper edge 
beveled outwards and we allow the water to flow over this. After the 
upper water had ceased to rise, the height of its surface above the bottom 
of the canal was 3|- feet and above the sill 1^ feet ; the theoretical dis- 
charge was therefore 

^ = 1 . 5 . 8,025 (Vf= 49,14 cubic feet. 

fi -i ft 

The coefficient of efflux is in this case, since — = -^ = |. and n^ = 0,577, 

a^ 0,0 

n, = [1,041 4- 0,3693 iff] . 0,577 = 1,110 . 0,577 = 0,64, 

and therefore the effective discharge is 

Q^ = 0,64 . Q = 0,64 . 49,14 = 31,45 cubic feet. 



84:6 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 418. 



§ 418. Lesbros's Experiments. — We are indebted to Mons. 
Lesbros for a great number of experiments upon the efflux of water 
through rectangular orifices in a thin plate; the orifices, being 
provided internally and externally with rims, afforded examples of 
both partial and incomplete contraction (see his " Experiences hy- 
drauliques sur les lois de Tecoulement de Feau"). We will give 
here only the principal results of his experiments with a rectangu- 
lar orifice 2 decimeters wide. The orifices, which were surrounded 
with borders of different kinds, are distinguished from each other 
in Fig. 715 by the letters A, B, C, etc. 



fiipji! 



i. I 



Fig. 715. 



B 




C 




D 



i 


\ 




F 



G 






A denotes the ordinary mouth-piece without any rim or border 

(as in § 410) ; 
B denotes a similar mouth-piece with a vertical wall upon the 

inside perpendicular to the plane of the orifice and at a distance 

of 2 centimeters from one side of it ; 
C denotes the first mouth-piece enclosed on the inside by two 

such walls ; 
D the orifice A, provided on the inside with two vertical walls, 

which converge towards each other at an angle of 90° and cut 

the plane of the orifice at an angle of 45° and at a distance of 

2 centimeters from the side of it ; 
E the orifice A with a horizontal wall, which passes across the 

reservoir and reaches exactly to the lower edge of the orifice ; 
F the orifice B, 
G the orifice C, and 
H the orifice D with a horizontal rim or wall, as in E, which 

completely prevents the contraction at the lower edge of the 

orifice. 



§ 418.] CONTRACTION OF THE VEIN OR JET OF WATER. 847 



TABLE OF THE COEFFICIENTS OF EFFLUX FOR FREE EFFLUX 
THROUGH THE ORIFICES A, B, G, ETC. 



i of water above 
upper edge of 
orifice, measured 
k from the plane 
he orifice. 


9J 

1 



Cm 


•1,1 


Coefficient of efflux for the orifices. 


















s 0) oj y -^ 




A 


B 


G 


D 


E 


F 


G 


H 


Meters. 


Meters. 
























0,020 


>V f 


0,572 


0,587 




0,589 


0,599 




— 




0,050 




0,585 


0,593 


0,631 


0,595 


0,608 


0,622 




0,636 


0,100 




0,592 


0,600 


0,631 


0,601 


0,615 


0,628 




0,639 


0,200 




0,598 


0,606 


0,632 


0,607 


0,621 


0,633 


0,708 


0,643 


0,500 


> 0,200 - 


0,603 


0,610 


0,631 


0,611 


0,623 


0,636 


0,680 


0,644 


1,000 




0,605 


0,611 


0,628 


0,612 


0,624 


0,637 


0,676 


0,642 


1,500 




0,602 


0,611 


0,627 


0,611 


0,624 


0,637 


0,672 


0,641 


2,000 




0,601 


0,610 


0,626 


0,611 


0,619 


0,636 


0,668 


0,640' 


3,000 


■^ v 


0,601 
io,6i6 


0,609 
0,627 


0,624 


0,610 


0,614 


0,634 


0,665 


0,638: 
0,678 


0,020 


1 r 


0,647 


0,631 


0,664 


0,663 




0,050 




0,625 


0,630 


0,646 


0,632 


0,667 


0,669 


0,690 


0,677 


0,100 




0,630 


0,633 


0,645 


0,633 


0,669 


0,674 


0,688 


0,677 


0,200 




0,631 


0,635 


0,642 


0,633 


0,670 


0,676 


0,687 


0,675 


0,500 


■ 0,050 ^ 


0,628 


0,634 


0,637 


0,632 


0,668 


0,676 


0,682 


0,671 


1,000 




0,625 


0,628 


0,635 


0,627 


0,666 


0,672 


0,680 


0,670 


1,500 




0,619 


0,622 


0,634 


0,621 


0,665 


0,670 


0,678 


0,670 


2,000 




0,613 


0,616 


0,634 


0,615 


0,664 


0,670 


0,674 


0,669 


3,000 


J V 


- 0,606 


0,609 


0,632 


0,608 


0,662 


0,669 


0,673 


0,668 



848 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 418. 



11. 

TABLE OF THE COEFFICIENTS OF EFFLUX THROUGH THE ORI- 
FICES A, B, G, ETC., 

With external shoots or uncovered canals of the same dimensions as 
the orifice (Fr. canaux de fuite ; Grer. aussere Ansatzgerinnen). 

The shoots fitted the orifice exactly, and consequently the bev- 
eling of the sides and bottom of the mouth-piece was done away 
with. They were either horizontal and 3 meters long or (in the 
experiments marked with *) inclined -^^ of their length, which was 
but 2,5 meters. 



above the 
"the orifice 
k from the 
rifice. 


QJ 


CoeflSdents of effltix for the orifices. 


Head of water 
upper edge ol 
measured bac 
plane of the c 


f 
■s 

.i 




A 


B 


G 


E 


E^ 


F 


F^ 


G 


6^* 


H 


Meters. 


Meters. 
















0,02 


\ r 


0,480 


0,489 


0,496 


0,480 


0,527 










0,488 


0,050 




0,511 


0.5 17 


0,531 


0,510 


0,553 


0,509 


0,546 


0,528 


— 


0,520 


0,100 




0,542 


0.545 


0.563 


0,538 


0,574 


0.534 


0,569 


0,560 


0,593 


0,552 


0,200 




0,574 


0,576 


0,591 


0,566 


0,592 


0,562 


0,589 


0,589 


0,617 


0,582 


0,500 


> 0,200 " 


0,599 


0,602 0,621 


0,592 


0:607 


0,591 


0,608 


0,591 


0,632 


0,613 


1,000 




0,601 


0,609 0,628 


0,600 


0,610 


0,601 


0,615 


0,601 


0,638 


0,623 


1,500 




0,601 


0,610 


0,627 


0,602 


0,610 


0,604 


0,617 


0,604 


0,641 


0,624 


2,000 




0,601 


0,610 


0,626 


0,602 


0,609 


0,604 


0,617 


0,604 


0,642 


0,624 


3.000 


^ ^ 


0,601 


0,609 


0,624 


0,601 


0,608 


0,602 


0,616 


0,602 


0,641 


0,622 


0,020 


-J f 


0,488 


0.555 


0,557 


0,487 


0,585 


0,483 


0,579 


0,512 




0,494 


0,050 




0,577 


0,600 


0,603 


0,571 


0,614 


0,570 


0,611 


0,582 


0,625 


0,577 


, 0,100 




0,624 


0,625 


0,628 


0,605 


0,632 


0,609 


0,628 


0,621 


0,639 


0,616 


0,200 




0,631 


""^^ZZ 


0,637 


0,617 


0,645 


0,623 


0,643 


0,637 


0,649 


0,629 


; 0,500 


> 0,050 " 


0,625 


0,630 


0,635 


0,626 


0,652 


0,630 


0,650 


0,647 


0,656 


0,636 


1,000 




0,624 


0,627 


0,635 


0,628 


0,651 


0,633 


0,651 


0,649 


0,656 


0,638 


1,500 




0,619 


0,622 


0,634 


0,627 


0,650 


0,632 


0,651 


0,647 


0,656 


0,637 


2,000 




0,613 


0,616 


0,634! 


0,623 


0,650 


0,631 


0,651 


0,644 


0,656 


0.635 


3,000 


v 


0,606 


0,609 


0,632 


0,618 


0,649 


0,628 


0,651 


0,639 


0,656 ( 


0.632 



§ 419.] CONTRACTION OF THE VEIN OR JET OF WATER. 849 

Example.— What is the discharge through an orifice 2 decimeters 

wide and 1 decimeter high, when the lower edge is 0,35 meters below the 

level of the water and upon a level with the bottom of the vessel, 1) for 

free efflux, and 2) for efflux through a short horizontal shoot? We have 

in this case the orifice E^ and the head of water above the upper edge is 

= 0,35 — 0,10 =0,25 meters. Table I gives, when the head is = 0,20 and 

the height of orifice = 0, 20, the coefficient of efflux ^ = 0,621, and, on 

the contrary, when the height of the orifice is = 0,05 meters, (i = 0,670 ; 

hence for the first case of the problem we can put 

0,621 + 0,670 
fi = 2 = 0,645. 

Table II gives, on the contrary, by interpolation, for a head of water 
0,25 meters above the upper edge of the orifice, the following values for ju. 
0,566 + ^ (0,592 - 0,566) = 0,570, and 
0,617 + ^ (0,626 - 0,617) = 0,619; 
hence in the second case we can put 

0,570 + 0,619 ^ , 
^ = -^ g- = ^'^^^- 

The cross-section of the orifice is 

F = ab = 0,20 . 0,10 = 0,020 square meters; 
the mean head of water is 

h = 0,350 - 0,050 = 0,300 meters; 
and, consequently, the theoretical discharge is 

Q = F ^2~gli = 0,02 V2 . 9,81 . 0,3 = 0,02 Vs^SSG 
= 0,02 . 2,425 = 0,0485 cubic meters. 
The effective discharge is in the first case 

Q^ =z fi^ Q = 0,645.0,0485= 0,0313 cubic meters, 
and, on the contrary, in the second case, i.e., when a shoot is added, 

Q = fi^ Q = 0,594 . 0,0485 = 0,0288 cubic meters. 
According to the formula //„ = (1 + 0,155 n) fx^ of § 414, we can put for 
efflux with partial contraction //„ = fx,-^ = (1 + 0,52) ^^ = 1,052 //„, since 
.| = 1. of the periphery of the orifice is surrounded by a border. But for 
such an orifice with complete contraction we have, according to Table I, 
page 831, //„ = 0,616 ; hence 

fi, = 1,052 . 0,616 = 0,648, 
and the discharge is 

Qj = ^, g = 0,648 . 0,0485 = 0,0314 cubic meters, 
I.E., a little greater than that obtained by employing Lesbros's table. 

§ 419. M. Lesbros has also experimented upon efflux through 
overfalls, employing the same orifices A, B, C, etc., but not allow- 
ing the water to rise to the upper edge of the orifice. The principal 
results of these experiments are to be found in the following tables. 
54 



.J 



GENERAL PRINCIPLES OF MECHANICS 



[^ 419. 



TABLE I. 

Table of the coefficients of efflux (5 ji) for free efflux through 

overfalls or notches. 



Head of water 






Coefficients of efflux for the orifices. 




above the sill, 
measured where 
the water is still. 














A 


B 


c 


D 


E 


F 


G 


Meter 
















0,015 


0,421 


0,450 


0,450 


0,441 


0,395 


0,371 


0,305 


0,020 


0,417 


0,446 


0,444 


0,437 


0,402 


0,379 


0,318 


0,030 


0,412 


0,437 


0,435 


0,430 


0,410 


0,388 


0,337 


0,040 


0,407 


0,430 


0,429 


0,424 


0,411 


0,394 


0,352 


0,050 


0,404 


0,425 


0,426 


0,419 


0,411 


0,398 


0,362 


0,070 


0,398 


0,416 


0,422 


0,412 


0,409 


0,402 


0,375 


0,100 


0,395 


0,409 


0,420 


0,405 


0,408 


0,405 


0,382 


0,150 


0,393 


0,406 


0,423 


0,403 


0,407 


0,407 


0,383 


0,200 


0,390 


0,402 


0,424 


0,403 


0,405 


0,408 


0,383 


0,250 


0,379 


0,396 


0,422 


0,401 


0,404 


0,407 


0,381 


0,300 


0,371 


0,390 


0,418 


0,398 


0,403 


0,406 


0,378 



TABLE IL 

TaUe of the coefficients of efflux (| ^i) for efflux through weirs with 

short shoots or open canals. 



Head of wa- 
ter above the 






Coefficients of efflux for the orifices. 






sill, measur- 
ed where 1 he 
water k- still. 

1 


















A 


B 


c 


D 


B 


F 


G 


B 


Meter. 


















0,015 




0,375 


0,388 


0,400 




— 




— 


0,020 


0,196 


0,368 


0.383 


0,395 


0,208 


0,201 


0,175 


0,190 


0,030 


0,234 


0,358 


0,373 


0,385 


0,232 


0,228 


0,205 


0,222 


0,040 


0,263 


0,351 


0,365 


0,379 


0,251 


0,250 


0,234 


0,250 


0,050 


0,278 


0,346 


0,360 


0,375 


0,268 


0,267 


0,260 


0,272 


0,070 


0,292 


0,343 


0,352 


0,371 


0,288 


0,289 


0,285 


296 


0,100 


0,304 


0,340 


0,345 


0,369 


0,302 


0,304 


0,299 


0,313 


0,150 


0,315 


0,335 


0,340 


0,367 


0,314 


0,316 


0,313 


0,327 


0,200 


0,319 


0,331 


0,338 


0,366 


0,323 


0,322 


0,322 


0,335 


0,250 


0,321 


0,328 


0,336 


0,364 


0,329 


0,326 


0,329 


0,341 


0,300 


0,324 


0,326 


0,334 


0,361 


0,332 


0,329 


0,332 


0,345 



A comparison of the coefficients m Table I and Table II shows 
that the discharge through orifices provided with shoots js smaller 
than that through those without them, and that the difference is 
greater, the smaller the head of water is ; we also see, by comparing 



^ 419.] CONTRACTION OF THE VEIN OR JET OF WATER. 851 

the columns C and C*, E and ^*, F and i^*, and G and G^ in the 
tables of the last paragraph, that the inclined shoot creates less dis- 
turbance in the efflux than the horizontal one. 

Remark 1.— A ciiiierent theory of the efflux of water is advanced by G. 
Boileau m his " Traite siir la mesure des eaux courantes." According- to 
it the velocity of the effluent water is the same at all parts of the cross-sec- 
tion and depends upon the depth of the upper limiting line of the vein at 
the plane of the orifice below the level of the water in the reservoir. 
Boileau employs the same formula for overfalls, in which case he must 
know of course the height of the stream in the plane of the orifice. Later, 
in the 12th volume of the 5th series of the Annales des Mines, 1857, M. 
Clarinval has given another formula for efiiux through overfalls in which no 
empirical number // appears, but instead cf f w he substitutes the factor 



, , in which li denotes the head of water and a the thickness of 

V27A^ - a")' 

the stream above the sill of the overfall. See the '' Civilingenieur," Yol. 

5th. I consider the hypothesis upon which this formula is based to be 

incorrect. 

Remabk 2. — Mr. J. B. Francis gives in his work " The Lowell Hydraulic 
Experiments, Boston, 1855," the following formula for efflux through a 
wide overfall or weir. 

Q = 3,33 {l — ^.lnh) Enghsh cubic feet, 
in which h denotes the head of water above the sill of the weir, I its length, 
and n either or 1 or 2, according as the contraction of the vein is pre- 
vented upon both, one or none of the sides. Since for the English system 
of measures 

VS^ = 8,025, 
we have 

^ ^ = 1^ = ^'^^^- 

The experiments, upon which this formula is based, were made with 
weirs 10 feet wide and under heads of water from 0,6 to 1,6 feet. The edge 
of the weir was formed of an iron plate beveled down stream, the reservoir 
was 13,96 feet wide, and the sill was 4,6 feet above its bottom. See the 
Civilingenieur, Vol. 2, 1856. 

Bakewell's experiments upon efflux through weirs or overfalls give 
results differing in some respects from the above. (See Polytech. Central 
Blatt, 18th year, 1852.) 

Remark 3. — At the sluice-gate of the wheel at Remscheid, Herr Ront- 
chen found // = 0,90 to 0,93. See Dingler's Journal, Vol. 158. 

A new edition of Mr. J. B. Francis' work has been recently published by 
B. Van Nostrand, New York.— [Tr.] 



S52 



GENERAL PRINCIPLES OF MECHANICS. 



[§420, 



CHAPTER III. 



OF THE FLOW OF WATER THROUGH PIPES. 

§ 420. Short Tubes. — If we allow the water to discharge 
through a short tube, or ^^ip^y called also an ajutage, (Fr. tuyau 
additionel ; Ger. kurze Ansatzrohre), the condition of affairs is 
entirely different from that existing, when the water issues from 
an orifice in a thin plate or from an orifice in thick wall, which is 
rounded off on the outside. If the short tube is prismatic and 2tI- 
to 3 times as long as wide, the stream is uncontracted and non- 
transparent and its range and consequently its velocity is smaller 
than when it issues, under the same circumstances, from an orifice 
in a thin plate. If, therefore, the tube K L has the same cross- 
section as the orifice F, Fig. 716, and if the head of water is the 

Fig. 716. 




game for both, we obtain at J? X a troubled and uncontracted op 
thicker stream and at i^ iT a clear and contracted or thinner one ; 

we can also see that the range E R 
is smaller than the range D H. 
This condition of efflux exists only 
when the length of the tube is the 
given one ; if the tube is shorter, 
E.G. as long as wide, the vein K R, 
Fig. 717, does not touch the sidss 
of the tube, the latter has then no 
influence upon the efflux, and the 
stream issues from it as from an 
orifice in a thin plate. 
Sometimes it happens, when the length of the tube is greater. 




§431.] THE FLOW OF WATER THROUGH PIPES. Sb^} 

that the stream does not fill it ; this occurs when the water has 
no opportunity of coming in contact with the sides of the tuhe ; 
if in this case we close for an instant the outside end of the tube 
with the hand or with a board, the stream will fill the tube and we 
have the so-called discharge of a filled tube (Fr. a gueule bee; 
Ger. voller Ausfluss). The vein is contracted in this case also, but 
the contracted portion is within the tube. We can satisfy our- 
selves of this by employing glass tubes like K L, Fig. 718, and by 

throwing small light bodies 
Fig. 718. jj^^q ^j-^q ^ater. U]oon so do- 

!z!!zfS ^T^ ing, we observe that near the 

^^ rS^ ^^ ^ i( y ; ^;;^::::::^ nr;:^ cntrancc K thcrc is a motion 

-^^^^ i> .--^ F ^.^§S^^ of translation in the middle of 

the cross-section F^, but that^ 
on the contrary, at the peri- 
phery of the same the water 
forms an eddy. It is, however, 
'^^' the capillarity or adhesion of 

the water to the walls of the tube, which causes it to fill the end F L 
of the tube completely. The pressure of the water discharging 
from the tube is that of the atmosphere, but the contracted cross- 
section i^i is only a times as great as that F of the tube ; the 

velocity Vx at that point is therefore - times as great as the velocity 

(Jb 

of efSux V and the pressure of the water at F^ is smaller than that 
at the end of the tube, which is equal to the pressure of the atmo- 
sphere. If we bore a small hole in the pipe near F^ no water will 
run out, but air will be sucked in and the discharge with a filled 
tube ceases, when the hole is enlarged or when several of them are 
made. We can also cause the water in the tube A B to rise and 
flow through the tube K L hj making it enter the latter at F■^. 
The discharge with a filled tube ceases for cylindrical tubes, when 
the head of water attains a certain magnitude (see § 439, Chap. IV). 

§ 421. Short Cylindrical Tubes. — Many experiments have 
been made upon the efflux of water through sliort cylindriccd tubes, 
but the results obtained differ quite sensibly from each other. It 
is particularly Bossut's coefficients of efflux which differ most from 
those of others by their smallness (0,785). The results of the ex- 
periments Michelotti with tubes Ih to 3 inches in diameter, under 
a head of water varying from 3 to 20 feet, gave as a mean vakio 



854 



GENERAL PRINCIPLES OF MECHANICS. 



[§421. 



ji = 0,813. The results of the experiments of Bidone, Eytelwein 
tind d'Aubuisson differ but little from those of the latter. But, 
iiGcording to the experiments of the author, we can adopt for short 
cylindrical tubes as a mean value fi = 0,815. Since we found this 
coefficient for an orifice in a thin plate = 0,615, it follows that, 
when the other circumstances are the same, |i| = 1,325 times as 
much water is discharged through a short pipe as through an ori- 
fice in a thin plate. These coefficients increase, when the diameter 
of the tube becomes greater and decrease a little, when the head 
of water or the yelocity of efflux increases. According to some 
experiments of the author's, made under heads varying from 0,23 
to 0,6 meters, we have for tubes 3 times as long as wide 



When the width is 


1 


2 


3 


4 centimeters. 


fX = 


0,843 


0,832 


0,821 


0,810 



According to this table the coefficients of efflux decrease sensi- 
bly as the width of the tube increases. In like manner Buff found 
with a tube 2,79 lines wide and 4,3 lines long that the coefficient 
of efflux increased gradually from 0,825 to 0,855, when the head 
of water decreased from 33 to 1^ inches. 

For the efflux of water through short 2^(^'^c(,ll6lopi2yedical tubes 
the author found the coefficient to be 0,819. 

If the short tube K L, Fig. 719, is partially surrou7ided dy a 
border or rim in the inside of the vessel, if, e.g., one of its sides 
is flush with the bottom C D of the vessel and if partial contrac- 
tion is thus produced, according to the experiments of the author, 
the coefficient of efflux is not sensibly increased, but the water 



Fig. 719. 



Fig. 720. 





■ii;i)lii]lB 



§422.] THE FLOW OF WATL?. THROUGH PIPES. 855 

moves with different velocities in different parts of the cross-sec- 
tion, viz., upon the side C more quickly than upon the opposite one. 
If the face of the tube is not in the surface of the plate but 
projects into the vessel, like E, F, G, Fig. 720, it is then called an 
interior short tuhe. If the face of the tube is at the least 5 times as 
wide as the bore of the tube, as at E, the coefficient of efflux remains 
the same as if the face were in the plane of the wall, but if the 
face of the tube is smaller, as at F and G, the coefficient of efflux 
is smallen According to the experiments of Bidone and of the 
author, if the face is very small, it is 0,71, when the stream fills the 
tube ; on the contrary, it is = 0,53 (compare § 113), when it does 
not touch the internal surface of the tube. In the first case (F) 
the stream is troubled and divergent like a broom, but in the 
second {G) it is compact and crystalline. 

§ 422. CoefiEicient of Resistance. — Since the stream of water 

issues from a short prismatical tube without being contracted, it 

follows that the coefficient of contraction of this mouth-piece a — 

unity and that its coefficient of velocity = its coefficient of efflux fx. 

The vis viva of a quantity of water Q, which issues with a velocity 

Qy v^ 

V, is — - v^ and its energy is x— Q y (see § 74). But the theoreti- 

9 ^9 

V 

cal velocity of efflux is -— , and therefore the theoretical energy of 

1 v' 
the water discharged is --^ • ^~ • Q 7- Hence the loss of energy 

of the quantity Q of water during the efflux is 

For efflux through orifices in a thin plate, the mean value of 
is 0,975 ; hence the loss of energy is 

[(wt^^-'^]fg^y = ''''^fg^y' 

for efflux through a short cylindrical pipe, on the contrary, </> == 
0,815, and the corresponding loss of energy is 

I.E., nearly 10 times as much as for efflux tlirough an orifice in a 
thin plate. Consequently if the vis viva of the water is to be made 
use of, it is better to allow it to flow through an orifice in a thin 
plate than through a short prismatical tubd. If, however, we 



LV 0,815/ 



856 



GENERAL PRINCIPLES OF MECHANICS. 



[§42^. 



round off the edge of the tube, where it is united to the interior 
surface of the vessel, so as to produce a gradual passage from the 
yessel into the tube, the coefficient of efflux is increased to 0,96 
and at the same time the loss of energy is reduced to 8 A per cent. 
For short tubes or ajutages, which are rounded off or shaped inter- 
nally like the contracted vein, we have fi = (f) = 0,975, and the 
loss of mechanical effect is the same as it is for an orifice in a thin 
plate, viz., 5 per cent. 



The loss of mechanical effect 



(^-•) 



V' 



Q y corresponds to a 



head of water ( —5 — 1 ) 7: — ; we can therefore consider that the loss 
\0 / ^ff 



0* / ^ ^ 

of head due to the resistance to efflux is 



■<'-') 



v 



^9 



and we can 



assume that, when this loss has been subtracted, the remaining por- 
tion of the head is employed m producing the velocity. 

"1 \ 2 

This loss z — [-75 — 1 ) ^— J which increases with the square 
of the velocity, is known as the height of resistance (Fr. hauteur 

de resistance ; Ger. Widerstandshohe) and the coefficient — ^ — 1, 

by which the head of water must be multiplied in order to obtain 
the height of resistance, is called the coefficient of resistance. Here- 
after we will denote this coefficient, which also gives the ratio of 
the height of resistance to the head of water, by C oi* the height 

of resistance itself by ^ = ^ . -— . By means of the formulas 

Z g 

^ = —. — 1 and 
0' 
1 

0: 




we can calculate from the coefficient 
of velocity the coefficient of resistance, 
or the latter from the former.. 

If the velocity of efflux v is the 
same, the head of water of an orifice 
K, Fig. 721, whose coefficient of resist- 



ance is 0, is h 



V 



and the head 



j^\immmmmi^^^mi(^ 



2 g r 

of water of the orifice L, through which 
the water flows with this theoretical 



433.] 



THE FLOW OF WATER THROUGH PIPES. 



857 



V' 



velocity, is hi = ^r~, consequently the first orifice must lie at a dis- 



below the second 



/I \ ^;' -y' 

tance IC L = z = h — hi = (—: — 1) -— =z ^ —- 

V 0' / 2 g 2 g 

one. This distance z is called the height of resistance. If they 

have the same cross-section F and there is no contraction at either 

orifice, the discharge Q = F v is the same for both. 

Example — 1) What is the discharge under a head of water of 3 feet 
through a tube 3 inches m diameter, whose coefficient of resistance is 
C = 0,4. Here 



<P = 



Vl,4 



= 0,845 ; hence 



V = 0,845 . 8,025 V3 = 11,745 feet ; 
F = {^y TV = 0,02183 square feet, 

and consequently the required discharge is 

Q = 0,02182 . 11,745 = 0,256 cubic feet. 
3) If a tube 3 inches wide discharges under a head of 2 feet 10 cubic 
feet of water in a minute, the coefficient of efflux or velocity is 
^10 1 



^ = 



FV2gh 60 . 0,02182 . 8,025 V 2 1,05 V 2 
the coefficient of resistance ^ = ( TTor^) "~ ^ = 1,208, 
and the loss of head, caused by the resistance of the tube, is 

0,0155 ('^y = 0,0187. 



= 0,673, 



s=Cii— = 1,208.:^ = 1,208 
2g ' 2g 



0,1309' 



1,092 feet. 



§ 423. Inclined Short Tubes or Ajutages.— When the 

tubes are applied to the vessel in an inclined position or when 

they are cut off obliquely to the axis, the discharge is less than 

when they are inserted, into the vessel at 

right angles or cut off at right angles to 

their axis ; for in this case the direction of 

the water is changed. The author's extended 

experiments upon this subject have led to 

the following conclusions. If S denotes the 

angle L K N, formed by the axis of the tube 

K L, Fig. 722, with the normal K N to the plane A B of the 

orifice, and if ^ denotes the coefficient of resistance for tubes cut 

off at right angles, we have for the coefficient of resistance of in- 

chned tubes 

^j = ^ + 0,303 si7i. S + 0,236 sin' ^. 

Assuming for ^ the mean A'alue 0,505, we obtain 




858 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 424. 



for 5° = 


O 


10 


20 


30 


40 


50 


60 deg. 


the coefficients of 
resistance ^i = 

the coefficient of 
efflux M-i = 


0,505 
0,815 


0,565 

0,799 


0,635 
0,782 


0,713 
0,764 


0,794 

0,747 


0,870 
0,731 


0,937 
0,719 



Hence, e.g., the coefl&cienfc of resistance of a short tube, the 
angle of deviation of whose axis is 20°, is ^i = 0,635 and the coeffi- 
cient of efflux is 

1 

= 0,782, 



^1 



|/l,635 



and, on the contrary, when the deviation is 35°, the former is 
=: 0,753 and the latter = 0,755. 

These inclined tubes are generally longer than those we have 
previously considered, and they must be longer Avhen they are to 
be completely filled with water. The foregoing formula gives only 
that part of the resistance due to the short tube at the inlet 
orifice, that is, three times as long as the tube is wide. The resist- 
ance of the remaining part of the tube will be given further on. 

Example. — If the plane of the orifice A B of the discharge-pipe K L^ 
Fig. 733, as well as the inside slope of the dam, is inclined at an angle of 40° 

to the horizon, the axis of the tube 
will form an angle of 50° with 
that plane ; hence the coefficient 
of resistance for efilux through 
the entrance of this pipe is C = 
0,870, and if the coefficient of re- 
sistance for the remaining longer 
portion is 0,650, we have the coefficient of resistance for the entire tube 

C = 0,870 + 0,650 = 1,520, 

and therefore the coefficient of efflux is 



Fig. 723. 




fJ- 



1 1__ 

Vl + 1,520 ^ V2;525 ^ ^'^^^• 

If the head of water is 10 feet and the width of the pipe 1 foot, the 
discharge is 



7r 



Q = 0,630 . J . 8,025 VlO 



12,56 cubic feet. 



§ 424. Imperfect Contraction. — If a short tube K L, Fig. 
724, is inserted in a plane wall, whose area G is but little larger 
than the cross-section i^of the tube, the water will approach the 



§ 424.] 



THE FLOW OF WATER THROUGH PIPES. 



859 



Fig. 724. 



mouth of the short tube with a velocity, which we cannot neglect, 
and the stream which enters it is imperfectly contracted ; hence 

the velocity of efflux is greater than 
when the water can be considered to 
be at rest at the mouth of the tube. 

F 

Now if 79" = 7^ is the ratio of the cross- 



section of the tube to that of the wall 
and 11^ the coefficient of efflux for perfect 
F 




'"'"""""rf:«!t(«n 



contraction, in which case we can put -— = 0, we have, according 

to the experiments of the author, for the coefficient of efflux with im- 
perfect contraction, when we put the ratio of the cross-sections = n, 



fJ'n — 1\ _ 



^0 



= 0,102 n + 0,067 n' + 0,046 n\ or 



11^ = 11^(1 + 0,102 n + 0,067 n' + 0,0^6 n'). 

If, E.G., we assume the cross-section of the tube to be one-sixth 
of that of the wall, we have 

li^ = li^{l + 0,102 . 1 + 0,067 . rh + 0,046 . -^\^) 

= ^„ (1 + 0,017 + 0,0019 + 0,0002) := 1,019 fi,, 

or putting fi^ = 0,815 

/x, = 0,815 . 1,019 -= 0,830. 

a 

The values — — ~ of the correction are given in the following 
tables, which are more convenient for use. 

TABLE OF THE CORRECTIONS OF THE COEFFICIENTS OF 
EFFLUX, ON ACCOUNT OF IMPERFECT CONTRACTION, FOR 
EFFLUX THROUGH SHORT CYLINDRICAL TUBES. 



n 


0,05 


0,10 


0,15 


0,20 
0,027 


0,25 
0,035 


0,30 


0,35 


0,40 
0,060 


0,45 
0,070 


0,50 


^0 


0,006 


0,013 


0,020 


0,043 


0,052 


0,080 







n 



^0 



0,55 



0,090 



0,60 



0,65 



0,102 



0,114 



0,70 0,75 



0,80 



0,85 



0,90 



0,95 1,00 



0,127 0,138 0,15210,166 0,1810,198 0,227 



860 GENERAL PRINCIPLES OF MECHANICS. [§424. 

When the water is discharged through short paraUelopipedical 
tubes, these corrections are about the same. 

The principal applications of these corrections are to the efflux 
of water through compound tubes, as, e.g., in the case represented 

in Fig. 725, where the short tube K L 
is inserted into another short tube 
' "^ G K, and the latter into the vessel 



D 



II .11 |l I 111' 




A C. Here, when the water enters 
^V^'v^\| the smaller from the larger tube, the 

vl^l^'^^^^^^^^s??^ stream is imperfectly contracted, and 

the coefficient of efflux is determined 
by the last rule. If we put the coef- 
ficient of resistance corresponding to this coefficient of efflux = C, 
the coefficient of resistance for its entrance into the larger tube 
from the reservoir = ^, the head of water = h, the velocity of 

F 
efflux = V and the ratio -^ of the cross-sections of the tube = n, 

vr 

or the velocity of the water in the larger tube = n v,we have the 
formula 

h = (1 + n^ ^ -{- ^i) ^ — , and therefore 

^ 2 / 

V2gh 



Example. — ^What is the discharge from the vessel represented in Fig. 
V25, when the head of water is A = 4 feet, the width of the narrow tube 3 
bches and that of the larger one 3 inches ? Here 

n = {%y = f, whence //, = 1,069 . 0,815 = 0,871 

9 

and the corresponding coefficient of resistance 

Ci = ((Tott) — 1 = 0,318; but we have 

C = 0,505 and Ti'' C = if . 0,505 = 0,099, 
whence it follows that 

l + ^'C+Ci = l-l- 0,099 + 0,318 = 1,417, 
and the velocity of efflux 

8,025 . V4 16,05 



V = 



Vl,417 Vl,417 



= 13,48. 



Finally, since the cross-section of the tube is i^ = TIi= 0,02183 square feet, 
it follows that the discharge is 

q = 13,48 . 0,02182 = 0,294 cubic feet. 



425.] 



THE FLOW OF WATER THROUGH PIPES. 



861 



Fig. 726. 




§ 425. Conical Short Tubes or Ajutages.— The discharges 
from conical mouth-pieces or short conical tubes are different froi» 
those obtained from cyhndrical or prismatic ones. They are either 
conically convergent or conically divergent. In the first case the 

outlet orifice is smaller than the 
inlet, and in the second case the 
inlet is smaller than the outlet. 
The coefficients of efflux through 
the former tubes are greater and 
those of efflux through the latter 
smaller than for cylindrical tubes. 
The same conical tube discharges 
more water when we make the 
wider end the orifice of discharge, as in K, Fig. 726, than when 
we put it in the wall of the reservoir, as is represented at L in the 
same figure ; but the ratio of the discharge is not as great as that 
of the openings. When authors such as B. Venturi and Eytelwein 
give greater coefficients of efflux for conically divergent than for 
conically convergent tubes, it must be remembered that the smaller 
cross-section is alwavs considered as the orifice. The influence of 
the conicalness of the tubes upon the discharge is shown by the 
following experiments, made under heads of from 0,25 to 3,3 
meters, with a tube A D, Fig. 727, 9 centimeters long. The width 

of this tube at one end was BE— 2,468, 
at the other A B = 3,228 centimeters, 
and the angle of convergence, i.e. the angle 
A B, formed by the prolongation of the 
opposite sides A E and B D of 2^^ section 
through the axis of the tube, was =40° 50'. 
When the water issued from the narrow opening, the coefficient of 
efflux was = 0,920 ; but when it issued from the wider opening, it 
was = 0,553. If we substitute in the calculation the narrower 
orifice as cross-section, we find it = 0,946. The stream, in the first 
case, when the tube was conically convergent, was but little con- 
tracted, dense and smooth ; in the second case, where the mouth- 
piece was conicaUy divergent, the stream was very divergent and 
torn and pulsated violently. Venturi and Eytelwein have experi- 
mented upon efflux through conically divergent tubes. Both these 
experimenters also attached to these conical tubes cylindrical and 
conical mouth -pieces, shaped like the contracted vein. With a 
compound mouth-piece, like the one represented in Fig. 728, the 



Fig. 727. 





862 GENERAL PRINCIPLES OB^ MECHANICS. [§426. 

diverging portion K L of. whicli was 12 lines in diameter in the 
iiarrowest place and 21} lines at the widest, and 8j| inches long, 
and whose angle of convergence was 5° 9', Eytelwein found f* = 
1,5526, when he treated the narrow end as the orifice, and, on the 
contrary, |Lt = 0,483 when, as was proper, he treated the larger end 

as the orifice. However, .^ -^ -- = 2,5 times as much 

0,515 

water is discharged through this compound mouth- 
piece as through a simple orifice in a thin plate, and 

1 5526 

^ = 1,9 times as much as through a short 

U,oXO 

cylindrical pipe. When the velocities and the angle of divergence 
are great, it is not possible to produce a complete efflux, even by at 
first closing the end of the moutli-piece. 

The author found with a short conically divergent mouth- 
piece 4 centimeters long, whose minimum and maximum widths 
were 1 and 1,54 centimeters and whose angle of divergence was 
8° 4', under a head of 0,4 meters, fi = 0,738 when the internal edge 
was rounded off, and fj^ = 0,395 when it was not. 

§ 426. The most extensive experiments upon the efflux of 
water through conically convergent tiibes are those made by d'Au- 
buisson and Castel. A great variety of tubes, which differed in 
length, width and in the angle of convergence, were employed. 
The most extensive were the experiments with tubes 1,55 centi- 
meters wide at the orifice of efflux and 2,6 times as long, i.e., 4 cen- 
timeters long ; for this reason we give their results in the follow- 
ing table. The head of water was always 3 meters. The discharge 
was measured by a gauged vessel, but in order to determine not 
only the coefficient of efflux, but also the coefficients of velocity 
and contraction, the ranges of the jet corresponding to the given 
heights were measured, and from them the velocities of efflux were 
calculated. 

The ratio ^ of the effective velocity v to the theoretical 

one ^Igli gave the coefficient of velocity 0, the ratio -^— ^ of 

the effective discharge Q to the theoretical discharge F V2 g h the 
coefficient of efflux fi, and, finally, the ratio of the two coefficients, 

I r. -, determined the coefficient of contraction a. 





427.] 



THE FLOW OF WATER THROUGH PIPES. 



863 



This determination is not accurate enough, when the velocities 
of efflux are great ; for in that case the resistance of the air is too 
great. 



TABLE OF THE COEFFICIENTS OF EFFLUX AND VELOCITY FOR 
EFFLUX THROUGH CONICALLY CONVERGENT TUBES. 



Angle of 
convergence. 


Coeflficient of 
efflux. 


Coefficient of 
velocity. 


Angle of 
convergence. 


Coefficient of 
efflux. 


Coefficient of 
velocity. 


o°o' 


0,829 


0,829 


13° 24' 


0,946 


0,963 


i°36^ 


0,866 


0,867 


14° 28' 


0,941 


0,966 


• 3° lo' 


0,895 


0,894 


16° 3^' 


0,938 


0,971 


4° lo' 


0,912 


0,910 


19° 28' 


0,924 


0,970 


5° 26^ 


0,924 


0,919 


/ 
21 


0,919 


0,972 


7° 52' 


0,930 


0,932 


23° 0' 


0,914 


0,974 


8° 58' 


0,934 


0,942 ' 


29° 58' 


0,895 


0,975 


10° 20' 


0,938 


0,951 


/ 
40 20 


0,870 


0,980 


12° 4' 

— — 


0,942 


0,955 


48° 50' 


0,847 


0,984 1 



According to this table, the coefficient of efflux attains its maxi- 
inum yalue 0,946 for a tube, whose sides converge at an angle of 13^-°,, 
that, on the contrary, the coefficients of velocity increase continu- 
ally with the angle of convergence. How the foregoing table is to 
be employed in practice, is shown by the following example. 

Example. — What is the discharge through a short conical mouth-piece 
t^ inches wide at the orifice of efflux and converging at an angle of 10°, when 
the head of water is 16 feet ? According to the author's experiments, a 
cylindrical tube of this width gives fu. = 0,810, d'Aubuisson tube, however, 
gave fj. = 0,829, or 0,829 — 0,810 — 0,019 more ; now, according to the 
table, for a tube converging at 10°, fi= 0,937 ; it is therefore better to put 
for the given tube ju = 0,937 — 0,019 = 0,918; whence we obtain the 
discharge 

Q = 0,918 . j^^ . 0,825 VI6 = ^~-— ^-?i^^-^ = 0,3616 cubic feet. 



4. S'' 



64 



§ 427. Resistance of Friction. — The longer prismatical or 
cylindrical pipes are, the greater is the diminution of the discharge 
through them ; we must therefore assume that the walls of the 
pipes by friction, adhesion or by the water's sticking to them resist 
the motion of the water. As we might suppose, and in accordance 
with many observations and measurements, we can assume that 



864 GENERAL PRINCIPLES OF MECHANICS. [§427. 

this resistance of friction is entirely independent of the pressure, 
that it is directly proportional to the length / and inversely to the 

diameter d of the pipe, i.e., it is proportional to the ratio -^. It has 

also been proved that this resistance is greater when the velocities 
are great and less when they are small, and that it increases, very 
nearly, with the square of the velocity v. If we measure this 
resistance by a column of water, which must afterwards be sub- 
tracted from the total head A, in order to obtain the height neces- 
sary to produce the velocity, we can put this height, which we will 
hereafter call the TieigM of resistance of friction, 

7, _ >• ^ ^' 

^ denoting here an empirical number, which we can style the co- 
efficient of friction. Hence the loss of head or of pressure in conse- 
quence of the friction of the water in the pipe is greater, the greater 

the ratio -^ of the length to the width and the greater the height 

due to the velocity — is. From the discharge Q and the cross- 
section of the tube 

we obtain the velocity 

4 Q 



T^ '^ d' 



ird'' 
and, therefore, the height of resistance of friction 

"' ~^' d' %g\r:dV ~^ ' 2g' Xn): d' ' 

If we wish to conduct a certain quantity Q of water through a 
pipe with as little loss of head or fall -as possible, we must make 
the pipe as short and as wide as we can. If the width of the pipe 
is double that of another, the friction in the former is (^)^ = 3 j 
that in the latter. 

If the cross-section of the pipe is a rectangle, whose height is a 
and whose width is hy we must substitute 

i — I ^^ _ 1 periphery _ ^ % {a -[- h) _ a -\- h 
d~ ""' \TTd'~ ""' area ~ ^ • ^ ^ - -Yob' 
whence we have 

2, _ >- ^ (^ + ^) ^'^ 
2a %g 



§ 428.] 



THE FLOW OF WATER THROUGH PIPES. 



865 



By the aid of these formulas for the resistance of friction in 
pipes, we can find the discharge and the velocity of efflux of the 
water conveyed by a pipe of a given length and width, under a 
given pressure. It is also of no consequence whether the tube K L, 
Fig. 729, is horizontal or inclined upwards or downwards, so long 

\ as v^^e understand by the 

Fig. 739. head of water the depth 

xr i^manteiw^ n R i? Z of the centre L of 

the mouth of the pipe 
below the level H oi 
the water in the reser- 
voir. 
If ti is the head of water, h^ the height of resistance for the ori- 
fice of influx, and y^g the height of resistance for the remaining part 
of the tube, we have 

h — {hi + h^) = X—, or A = ^ \- hi + h. 

If ^0 denotes the coefficient of resistance for the orifice of influx 
and ^ the coefficient of resistance of friction of the rest of the tube, 
we can put 




7, — '^^ j_ r ^' r ^ ^^ 



or 



and 



i)A=(i + ^. + 4)i: 

V2gli 



? 



2) v = 



/ 



1 + ^0 + ^ 



l_ 

d 



From the latter formula we obtain the discharge Q — Fv. 

For very long tubes 1 + ^^ is very small, compared with ^ ^, 
and we can write more simply 

A = 4" ^ • ^r— J or inversely, 
d % g -^ 



v=\/\. 



d 



C I 



.2gh, 



g 428. The coefficient of friction, like the coefficient of efflux, 
is not perfectly constant ; it is greater for low velocities than for 
high ones, I.E. the resistance of friction of the water in tubes does 
not increase exactly with the square, but with another power of the 

55 



S6Q GENERAL PRINCIPLES OF MECHANICS. [§428. 

yelocities. Prony and Eytelwein have assumed that the head lost 
by the resistance of friction increases with the simple velocity and 
with the square of the same, and have established for it the formula 

h =z {a V + 13 v^) -., 

in which a and ^ denote constants determined by experiment. In 
order to determine these constants, these authors availed themselves 
of 51 experiments made at different times by Couplet, Bossut, and 
du Buat upon the flow of water through long tubes. Prony de- 
duced from them 

li = (0,0000693 V + 0,0013932 v') -, 

Cv 

Eytelwein, 

h = (0,0000894 V + 0,0011213 v') -, 
d'Aubuisson assumes 

h = (0,0000753 V + 0,001370 v') ^ meters. 

The following formula, proposed by the author, coincides better 
with the results of observation ; it is 



V i/^J d 



V 

\/yl d 2 g 

and is founded upon the assumption that the resistance of friction 
increases at the same time with the square and with the square 
root of the cube of the velocity. We have, therefore, for the coeffi- 
cient of resistance 

Vv 
and for the height of resistance of friction simply 

^^-^'d^g 

For the determination of the coefficient of resistance ^ or of the 
auxiliary constants a and j3 the author availed himself of not only 
the 51 experiments of Couplet, Bossut, and du Buat, employed by 
Prony and Eytelwein, but also of 11 experiments made by himself 
and one by a M. Gueymard, of Grenoble. The older experiments 
were made with velocities of from 0,043 to 1,930 meters, but by the 
experiments of the author this limit has been extended to 4,648 
meters. The widths of the pipes m the older experiments were 
27, 36, 54, 135, and 490 milhmeters, and the newer experiments 



§428.] 



THE FLOW OF WATER THROUGH PIPES. 



867 



were made with pipes 33, 71, and 275 millimeters in diameter. By 
the aid of the method of least squares, the author found from the 
63 experiments 



^ = 0,01439 -f 



or 



0,00947 11 

Vv 

0,0094711\ I 



7. /r^Ai^on , 0,0094711\ I V' 

h = 1 0,01439 + : -— — I ^ . TT- meters, 

\ ^y / a 2g 

or for the Enghsli system of measure 



(o. 



h --= 0,01439 + 



0,017155\ I 



L7155\ l_ v^ 



Remakk — 1) If we take into consideration some other experiments made 
by Professor Zeuner with a zinc tube 25- centimeters wide, and with a ve- 
locity of from 0,1356 to 0,4287 meters, we obtain 



C = 0,014312 + 



0,010327 

■\/v 



v being given in meters. 

2) Newer experiments upon the flow of water with great and very great 
velocities were made by the author in 1856 and 1858 (see the " Civilinge- 
nieur," Vol. V, Nos. 1 and 3, as well as Vol. IX, No. 1). The results of 
these experiments are contained in the following table : 



Nature of the tubes. 



Narrow glass tubes 

Wider glass tubes 

Narrow brass tubes 

The same made shorter .... 
The same under very great pressure 
Wider brass tubes .."... 
The same made shorter .... 
The same under very great pressure 

Wider zinc tubes 

The same shorter . , . . . . 
The same still shorter .... 
The same still shorter .... 



Width of the 
tubes {d). 



1,08 ctm. 

1,43 

1,04 

1,04 

1,04 

1,48 

1,43 

1,43 

2,47 

2,47 

2,47 

2,47 



u 
u 
u 
u 
a 
(I 
u 
(( 
a 



Mean velocity of 
the water in the 
tubes {v). 



8,51 meters. 
10,18 " 

8,64 
12,82 
20,99 

8,66 
12,40 
21,59 

3,19 

4,73 

6,24 

9,18 



u 
u 
(( 
(( 
a 

(C 

u 
a 
a 
u 



Coefficient 
of friction ^. 



0,01815 
0,01865 
0,01869 
0,01784 
0,01690 
0,01719 
0,01736 
0,01478 
0,01962 
0,01838 
0,01790 
0,01670 



868 GENERAL PRINCIPLES OF MECHANICS. [§429. 

The values in the last column again show that the coefficient of resist- 
ance C, for the friction of water in tubes decreases not only as the velocity 
(«) increases, but also, although more slowly, as the width {d) of the pipe 
becomes greater. However, for high velocities, the formula 

^_,_ 0,0094711 
; = 0,01439 + ' 

agrees tolerably well with the numbers found by experiment, e.g., for 

2j = 9 meters 

C = 0,01439 + 0,00316 = 0,01755 

and for «> = 16 meters 

C = 0,01439 + 0,00337 = 0,01676. 

These coincide very well with the values in the last table, which corre^ 

spond most nearly to them. 

Remark 3. — M. de Saint -Venant found that the well-known formula 
for the resistance of water in tubes agrees better with the results of experi- 
ment, when we assume the height due to the friction to increase not with 

v"^ or ;;— , but with v^i. (See his " Memoire sur des formules nouvelles pour 

la solution des problemes relatifs aux eaux courantes.") According to him 
we must put 

47 7 I '1)^ 

^ = — . 0,00029557 t;¥ = 0,00118228 -, . «V = 0,023197 «-?.;, ^. 

Cb (Ju Ct lii Cf 

The assumption of a fractional exponent for i) is not at all new ; Woltmann 
put v\ mstead of ■y^ and Eytelwein proposed vU instead of tf (see the 
author's article upon Efflux [AusflussJ in the " allgemeine Maschinenency- 
clopadie " of Hiilsse. 

Remark 4. — New and very extended experiments upon the motion of 
water in pipes have been made by Monsieur H. Darcy (see the report to 
the Academy of Sciences at Paris in the Comptes rendus, etc., Tom. 38, 
1854, " sur des recherches ezperimentales relatives au mouvement des 
eaux dans les tuyaux "). Mons. Darcy deduces from these experiments, 
where the velocity is not less than 2 decimeters, the formula 

k = (0,00«507 + '^^^m^l\ I . ,. 
\ r I r 

/nn-rnon 0,0005078\ I v"" ^ 

= (0,01989+ ^ j^ 2^ meters; 

hence the coefficient of resistance should be 
f = 0,01989 + -^"I?. 
This formula, however, is not sufficiently accurate for small velocities.. 

§ 429. To facilitate calculation the following table of the 
coefficients of resistance has been arranged. We see from it that the 
variation of this coefficient is not insignificant, since for a velocity 



430.J 



THE FLOW OF WATER THROUGH PIPES. 



869 



of 0,1 meter it is = 0,0443, for one of 1 meter, = 0,0239 and for 
one of 5 meters, = 0,0186. 

TABLE OF THE COEFFICIENTS OF FRICTION OP WATER. 





Decimeters. 




V 


1 
2 
3 

4 





1 

0,0443 
0,0234 
0,0209 
0,0198 
0,0191 


2 


3 

0,0317 
0,0227 
0,0206 
0,0196 
0.0190 


4 


5 


6 


7 


8 

0,0250 
0,0215 
0,0201 
0,0193 

0,0187 


9 


00 

0,0239 
0,0211 
0,0199 
0,0191 


0,0356 
0,0230 

0,0208 
0,0197 
0,0190 


0,0294 
0,0224 
0,0205 
0,0195 
0,0189 


0,0278 
0,0221 
0,0204 
0,0195 
0,0189 


0,0266 
0,0219 
0,0203 
0,0194 
0,0188 


0,0257 
0,0217 
0,0202 
0,0193 
0,0188 


0,0244 
0,0213 
0,0200 
0,0192 
0,0187 



We find in this table the coefficients of resistance correspond- 
ing to a certain velocity by searching for the whole meters in the 
vertical columns and for the tenths of a meter in the horizontal 
column and then moving horizontally from the first number and 
vertically from the last, until we arrive at the point where the two 
motions meet. e.g. for v = 1,3 meters, ^ = 0,0227 ; for ?; =: 2,8, 
^ = 0,0201. 

For the English foot we can put 



V 


0,1 
0,0686 


0,2 


0,3 


0,4 


0,5 


0,6 


0,7 


0,8 


0,9 


C 


; 0,0527 


0,0457 


0,0415 


0,0387 


0,0365 


0,0349 


0,0336 


0,0325 


V 


1 


n 


n 


2 


3 


4 


6 


8 


12 
0,0193 


20 

0,0182 


0,0315 ( 


),0297 


0,0284 


0,0265 


0,0243 


0,0230 


0,0214 


0,020 


5 



Remark. — A more extensive and more convenient table is to be found 
in the Ingenieur, jDages 442 and 443. 

\ 

V g 430. Long Pipes . — In considering the motion of water in 
long pipes or combinations of pipes, the three principal questions 
to be solved are the following. 

1) The length I and the width d of the pipe and the quantity 
Q of water to be conducted may be given and we may be required 
to find the necessary head. In this case we must first calculate 
the velocity 



870 GENERAL PRINCIPLES OF MECHANICS. [§430. 

and then search, in one of the last tables for the value of the coef- 
ficient of friction ^, corresponding to this value, and finally we 
must substitute the values d, I, v, ^ and ^^ {^^ denoting the coefii- 
cient for the orifice of influx) in the first principal formula 

2) The length and width of the pipe and the head of water 
may be given and the discharge may be required. The velocity 
must be found by means of the formula 

V¥gli 



V — 



l/l + ^o+^.^ 



d 

Now as the coefficient of resistance is not perfectly constant, 
but varies somewhat with v, we must first find v approximatively 
in order to be able to calculate ^ from it. 

From V we determine 

Q = '^v ^ 0,7854 d' v. 

3) The discharge, the head of water and the length of the pipe 
may be given, and we may be required to determine the necessary 
width of the pipe. 

4 
Since v — — -p or y^ = ( — ^ ) . -^, we have 



3W* = (l + C + 4,) (";?/.]., or 
%gn.{£-^'d' = (1 + Qd + a; 



hence the width of the pipe is 



^ g h \ 71 / 

But since ( ) = 1,6212 and 1 + ^o as a mean = 1,505 and for 

the English system of measures -x — = 0,0155, we can put 

d = 0,4787 1/(1,505 .d + ^l)^ feet. 
This formula can only be used to obtain approximative values ; 



I430.J THE FLOW OF WATER THROUGH PIPES. 871 

for not only tlie unknown quantity d, but also the coefficient ^, 

4 Q 
which depends upon the yelocity v = --^g, occurs in it. 

Example 1) What must the head of water be, when a set of pipes 150 

feet long and 5 inches in diameter is required to dehver 25 cubic feet of 

water per minute ? Here we have 

25 12^^ 
v = 1,2732 ^' ^, = 3,056 feet, 
bO . 5 

and therefore we can make C = 0,0243 ; hence the head of water or total 
fall of the pipes must be 

(150 12\ 
1,505 + 0,0243 . ^ — j . 0,0155 . 3,056' 

= (1,505 + 8,748) 0,0155 . 9,339 = 1,484 feet. 
2) What is the discharge through a set of pipes 48 feet long and 3 
inches in diameter, under a head of 5 feet ? Here 

8,025 V5 17,945 



V = 



1/ 



1,505 + C.^V-' ^'''"' + '''-^ 



For the present, assuming C = 0,020, we obtain 

17,945 17,945 

V = -^= = -i-w- = 6,6 ; 

V7,26 2,7 ' ' 

"but V = 6,6 gives more correctly C = 0,0211, and therefore we have 

17,945 17,945 ^ ko f ^ 

V = — ' =: — - — = 6,52 feet, 

Vl,505 + 288 . 0,0211 V7,582 
and the discharge 

Q = 0,7854 (^ 6,52 = 0,142 cubic feet = 245,4 cubic inches. 

3) What must be the diameter of a set of pipes 100 feet long, which are 
to discharge one half of one cubic foot of water per second under a head 
of 5 feet ? Here 



d = 0,4787 V (1,505 ^ + 100 C) . i . (|)' = 0,4787 V0,075 d + 5 C, 
Assuming for the present C = 0,02, we obtain 

d = 0,4787 V 0,075 ^^+"0,100, or approximatively 

d = 0,4787 V 0,100 = 0,30 ; hence we have more accurately 

d = 0,4787 V0,0225 + 0,100 = 0,4787 Voa325 
= 0,3145 feet = 3,774 inches. 
This diameter corresponds to tbe cross-section 
F = 0,7854 . 0,3145^ = 0,0777 square feet ; 
the velocity is consequently 
Q 0,5 

^ ^ Y ^ 00^777 ~ ^' ^^^^' 
and the coefficient of resistance C = 0,212. Substituting the latter nk *« 
correct value, w^e obtain 

d = 0,4787 n/0J285 = 0,318 feet = 3,82 inches. 



872 GENERAL PRINCIPLES OF MECHANICS. [§ 4ai; 

Remark 1. — Experiments made by the author with ordinary wooden, 
pipes 2^ and 4^ inches in diameter gave coefficients of resistance 1,75 times 
greater than those for metal pipes, given in the tables in the foregoing par- 
agraph. While we have, when the velocity is 3 feet, for metal pipes C ~ 
0,0243, for wooden pipes its value is = 0,0243 . 1,75 . 0,042525 ; in example 1 
we found for a metal pipe 150 feet long the head to be 1,484 feet, but for a 
wooden pipe under the same circumstances it would be 
h = (1,505 + 0,042525 . 360) 0,0155 . 9,339 = 16,81 . 0,1448 = 2,43 feet. 

According to D'Arcy's Experiments, the coefficient of resistance C in- 
creases very considerably with the roughness of the walls of the pipe, and 
if the walls are very rough it is doubled or even trebled. The author 
found more recently the same result. 

Remark 2. — The temperature also has an important influence upon the 
resistance of water in pipes. Experiments have been made upon this sub- 
ject by Gerstner (see his " Handbuch der Mechanic," Vol. II), and more 
recently by Geh. Rath Hagen (see his " Abhandlungen iiber den Einfluss 
der Temperatur auf die Bewegung des Wassers in Rohren," Berlin, 1854). 
The experiments of the latter, made, it is true, with very narrow tubes 
{d = 0,108 to 0,227 inches), have shown that under the same circumstances 
the velocity of the water in pipes does not decrease indefinitely with the 
temperature, but that for every tube there is a certain temperature for 
which this velocity is a maximum. For the experiments without this 
maximum, Hagen finds the following formula : 
h = m I r"^'*^ . v^^''^, and 

m = 0,000038941 - 0,0000017185 VI;, 
in which the temperature t is expressed in degrees of the Reaumur ther- 
mometer, and the head h, the length Z, the radius of the tube r and the 
velocity v in inches (Prussian). 

(§ 431.) Conical Pipes.— The resistance of friction in a conical 

pipe A B, Fig. 730, can be found in the following manner. Let us 

denote the semi-angle of convergence of the walls of the 

Fig. 730. pjpe A CL = B O Lhj6, the diameter of the inlet 

C orifice by d^, that of the outlet by di, the length K B 

M of the pipe by l, and the velocity of efflux at B Bhj v. 

I i \ At a distance K M~ x from the outlet of the tube 

/ I \ the diameter of the tube is 

tt/ IT \d JV0 = y = BB+2 K Mtang. 6 = d^_ -\- 2x tang. 6, 

fl hence for the velocity to at that point, since 
lo — — ~^ii we can put 



§431.] THE FLOW OF WATER THROUGH PIPES. 873 

For an element N P R of the tube, whose length is 

OP = NR = ^-^^ = -^^, 
COS. o cos.o 

the height of resistance of the friction is 

/7 /y ti'!^ 



dri — C, , 



d X w^ _ ^ d X v^ 

y COS. d ' %g ~^' .(. 2:e ^ Ty * ^ 

^ y COS. (5(1+ — tang. 6\ ^ 



d X V 



tta C0<§. O 1 1 + -^ ^6??Z^. I 

hence the height of resistance of friction for the whole tube is 
7, _ r '^'^ /*' ^ a; 

^ ^ ° (1 + -^ ^«w^. (5) C05. cJ 

But 
/* dx 

(1 H- -^ ^«^^. (j) COS. (5 

= ¥^sSk + ¥ ^''»^- '') '^^ (If ^"''^- ^) ' 

d I 2 X \~* 

= — Q — A-T (l + -7- tang. 6j , whence we obtain 
o sin. \ CC2 ' 

r^ d_x 

f 1 + -T- tang. 6\ cos. 6 

= ^' h - ^^X'^ = ^' It - i^\^ 

^ sin. SI \dj J 8 sin. 61 \ dj J 

since d<i + 2 I tang. 6 exjDresses the diameter di of the inlet orifice. 
Consequently the required height of resistance is 

h = C. — - . ^' [1 - (— VT 
' 2 g d^' S sin. dL \ ^1 / J 

^ siTi. c5 L \dj .\ 2 g ^ L ^dj A2 g 

If the inlet orifice is much larger than the outlet orifice, we can 

/ d.2 \' 
put \-j 1 = 0, and consequently 



g — — J • i^— = s s^ cosec. . ^r—] 
^ sm. 6 2 g ^ 2 g^ 



h = lS 



874 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 43S. 



the resistance of friction in this case does not depend at all upon 
the length of the tube. 

Example. — If the angle of convergence of the outlet portion of the 
nozzle A K, Fig. 731, of a fire-engine is 2 c5 = 5°, that of the inlet portion 
A B^2 6^ = 18°, the width of the outlet d^ =1 lines, and the width of the 
inlet d^ = 1^ inches = 18 lines, and if its whole length AK=l = Q inches 
= 72 lines, what is its coefficient of resistance ? Putting the length of the 
outlet portion B K = l^ and that of the inlet portion A B = Zg, we have 

I =z l^ + I and Zj tang. 6 + l^ tang, d^ = ^ — -^ 

or in figures 

Zj + Zg = 72 and Z^ tang. 2|-° + Zg tang. 9° = -y-, or 

0,04362 Zj + 0,15838 Zg = 5,5. 

Hence Zj = 51,54 and l^ — 20,46 lines and the width at B, 

where the conical surfaces meet each other, is 

6Z3 = 6Z3 + 2 Zi tang. (5 = 7 + 2. 51,54 . 0,04362 = 11,53 lines. 

Since this place is rounded off, we can put d^ = 13 lines; 

hence for the outlet piece 



Fig. 781 

Ci 



[' - m 



sm. 



(tV)1 



cosec. 2|-°. 




and for the inlet portion 



= 0,9159 . 22,926 = 21,08, 



6)1 



^ I cosec. d^ = [1 — (f|)*] . cosec. 9° 

= 0,7795 , 6,392 = 4,98. 
Therefore the heierht of resistance for the entire nozzle is 



h 



g 



= |[.M8 + 4,08.(1)*].^ 

= 3|21,08 + 4,98. yj^= 21,5. |.- 

= 2,7 r 



27' 

if we substitute - — = 0,0155 and assume C = 0,02, we have 

2g ' ' ' 

h = 0,054 . -— , 

2g 

I.E. about ^V^^^beight due to the velocity, which result coincides very well 
with the results of experiments with such a nozzle. 

§ 432. Conduit Pipes. — The outlet at the end of a system 
of pipes is either under water or in the air. Both cases are repre- 
sented in Figures 732 and 733. In the first case we must regard 
as the head h the difference of level R C o^ the two surfaces of 
water, and in the second case the vertical distance R of the out- 
let orifice below the level H of the water in the reservoir. If the 



432.] 



THE FLOW OF WATER THROUGH PIPES. 



875 



tube is everywhere of the same width d, the formulas found in 
§ 430 can be apphed directly \ but if the tube is enlarged or nar- 



FiG. 732. 



Fig. 733. 



ha_ 




rowed at any point, we will have several different velocities in the 
pipe, and therefore the resistance of friction for each portion of 
the pipe must be calculated separately. Such a case is presented 
by the pipes in Fig. 733, which lead to a fountain or jet d'eau, in 
which case the mouth-piece is narrower than the pipe B L M, 
which conveys the water. If we put, as we generally do, the ve- 
locity of efflux = V, the width of the orifice of efflux = d, the 
width of the pipe = d^, we have the velocity of the water in the pipe 



"• = (1)'"' 



and if we denote by I the length of the pipe B L M and by C, the 
coefficient of friction, we have for the corresponding height of 



friction 



Ai = ^1 



l^v;^ 



Ci 



I. 



\ \dj 






dx 2 g ^' dx 

Now if ^0 is the coefficient of friction for the inlet orifice K and 
^ that for the outlet orifice 0, it follows that the loss of head caused 
by the first ^^ ■, _ ^ v^^ _ ^ I d V v"" 

^'~^'2^~^'\dJ'2^' ■ 
and, on the contrary, that occasioned by passing through the 
second is , v ^^ 

hence we have the entire head 



h = 



V 



^ff 



+ A. + A, + A, = [l + ?, (|)V ^, ^ (|)'+ ?] ^, 



and inversely the velocity of efflux 



2gh 



1 + 



(<■ - <■ ^) (i)' 



+ ^ 



If we wish the jet to rise to the greatest height, the orifice or 
mouthpiece must not only cause as little resistance as possible, but 
also allow the water to issue from it with its fibres nearly parallel, 
so that they may form, while rising, a stream which will hold to- 



876 



GENERAL PRINCIPLES OF MECHANICS. 



[§433. 



gether as long as possible, and consequently be less disturbed by 
the air than a stream which was more or less torn when it left the 
orifice. For this reason we prefer a short, cylindrical or slightly 
conical mouth-piece, with the orifice of influx rounded off, to an 
orifice in a thin plate or to the orifices of the form of the con- 
tracted stream, although the former cause a greater loss of velocity 
than the latter. The nodes and bulges, which a stream which has 
passed through the latter orifices forms or tends to form, give tlie 
air a much better chance to penetrate it than a cylindrical stream. 

§ 433. Jets of Water. — So long as the stream K L N, which 
flows vertically downwards through a horizontal orifice K, Fig. 734, 

remains continuous and is not broken up 
by the air, its cross-section L decreases 
more and more as the distance K L = x 
from the orifice increases. If c is the ve- 
locity of efflux and v the velocity at X, we 

have 

v"^ =^ 2 g X -\- c^ , 

denoting by F the cross-section of the ori- 
fice of efflux and by Y that of the stream 
at L, we have the following equation 

Fc= YvoyF' c' = Y' v\ 
from which we deduce the equation 

Y"" (c" -{-2gx) = Fc\ov 
F" & 




& -\-'^gx 

for the form of the cataract of Newton (see 
Newton's Principia Philosophise, Vol. II, 
Sect. VII). If the cross-section of the 
orifice JT is a circle, whose diameter is c?, 
the cross-section at L forms a circle, whose 
diameter is y and for which we can put 

t = -:r—^ — , or 



N 



y 



d' + 2g x' 
d 



V 



1 4- 



2 gx 



Experiments upon the internal consti- 
tution of falling streams of water have 



§433.] 



THE FLOW OF WATER THROUGH PIPES. 



877 



been made by Savart. See Poggendorff's Annalen der Physik, 
Vol. 33. 

The cross-section of a stream M 8, which rises vertically 
from a horizontal orifice M, increases gradually with its distance 
3£ = X from the orifice M; for here the velocity of the water 

at is 

V = Vc^ — 2 g X, and therefore 

c' — 2gx' 
hence we have for the diameter of the cross- section at 



f = 



& d' 



d 



c' — 2g X 



.ony 



V^ 



2 g X 



Denoting the height due to the velocity ^r— by h, we have sim- 



ply and generally 
' 2/i = 



d 



This formula becomes incorrect at its limits ; according to it, 
E.G. in the rising stream for x = h or Sit the apex S, the diameter 
of the stream would be 

d d 



y = v 



00. 



vi-zri 

This, however, is not the case ; for the various fibres of water, 
of which the stream is composed, are not really at rest at the 
highest point, but possess a small velocity radially outwards. If 

the stream of water 
J^<^''^^^' A C, Fig. 735, is in- 

clined to the horizon, 

this formula 
d 

y= -, ■ 







1 


/ ^ 


1 






// 




M ^S 




/ 





\ 


^ 








\ 






N 1 


\ 




M 


. 











V 



1 ± 



X 

h 



is still applicable, when 
we substitute instead 
of X the vertical projec- 
tion iV of the stream 
A 0. If the jet flows 



878 GENERAL PRINCIPLES OF MECHANICS. [§484 

out of the orifice at an angle v to the horizon, its maximum height" 

B Gis 

c^ (sin, vV , / . v., , 
a = — -^ = h (sm. vy (see § 39). 

Therefore its diameter (at the vertex C) is 

d d d 



2/ = 



. A a ^1 — {sin. vY Vcos. v 



In the descending portion CD of the stream, y becomes gradually 
smaller and smaller, and when the stream reaches the horizontal 
plane A Z), from which it started, y becomes again = d, if the air 
has produced no disturbance in the motion of the stream. 

§ 434. The height s, to which a yeftical jet of water will rise, 

is approximatively equal to height due to the yelocity h = - — , only 

when the yelocity of efi9.ux (c) is small. From the experiments 
made by the author (see the experiments upon the height of rise 
of jets of water with different mouth-pieces in the 5th yol. of the 
Zeitschrift des Vereins deutscher Ingenieure), the following facts 
concerning jets of water were ascertained. 

1) The resistance of the air for small yelocities of eflfl.ux, yiz., 
from 5 to 20 feet, or for heights of rise of from 1 to 6 feet, is so 

small that the height of rise of the jet may in this case without 

ff 

appreciable error be put equal to the height due to the yelocity — — . 

^^ 

2) If the height due to the yelocity does not exceed 75 feet or 

the yelocity of efflux 56 feet, the ratio of the height of rise to the 
height due to the yelocity can be expressed by the formula 

s 1 



.2? 



h a + (3 h + yh' 

in which a, (3 and y denote empirical coefficients to be determined 
for each mouth-piece. 

3) For jets, which issue from orifices in a thin plate, the con- 
stant a can be put = 1 ; hence we can assume that the resistance 
during the passage through the orifice as almost null, when the 
velocities are small, and that it is measurable only when the 
velocities are great. The coefficient of resistance for these orifices 
is therefore not constant, but increases from zero gradually with 



§484.J THE FLOW OF WATER THROUGH PIPES. 879 

the velocity ; the value ^ = 0,97, given in § 408, can only be con- 
sidered as a mean one. 

4) For the same velocity of efflux the height of rise increases 
with the thickness of the stream, or with the width of the orifice ; 
consequently the resistance of the air is smaller for thick than for 
thin streams. The height of rise increases, therefore, not only with 
the head, but also with the thickness of the stream. 

5) Under the same circumstance a stream, issuing from a circu- 
lar orifice, rises higher than one discharged from an aperture of a 
different shape (square, etc.) 

6) If the velocities of efflux and the widths of the orifices are 
the same, those streams which are not contracted rise higher than 
those which are, not only because the former are thinner, but also 
because the latter, in consequence of their contractions and expan- 
sions, oppose less resistance to the penetration of the air. 

If the other circumstances and relations are the same and if the 
velocities of efflux are not very small, the jets issuing from short 
cone-shaped and longer conical tubes or ajutages with an internal 
rounding off" attain the greatest height. 

Mariotte concluded from his experiments upon the height of 
rise of jets of water (see Meining's Translation of Mariotte's Prin- 
ciples of Hydrostatics and Hydraulics) with orifices in a thin plate 
4 to 6 lines in diameter and under heads of from 5 ^ to 35 feet that 
the head or height due to the velocity, necessary to produce the rise 
s, must be 



whence 



s 
h = s + --— Paris feet, 
o\)\j 



- = 1+^ = 1 + 0,003333 s. 
S oyjO 



The very extensive and varied experiments of the author, made 
under heads of from 3 to 70 feet, give, on the contrary, for circular 
orifices m a thin plate, when their diameter was 

1) 1 centimeter 

- = 1 + 0,0035305 h + 0,00005406 h\ and when it was 
s 

2) 1,41 centimeters 

- 3- 1 + 0,00237191 Ji + 0,00005609 h\ 

s 

h being given m English feet. 



880 



GENERAL PRINCIPLES OF MECHANICS. 



[§434 



With 

Fig. 736 
I 



a conical mouth-piece ABC, Fig. 736, 15 centimeters 

long and 1 centimeter wide at the outlet 

Fig. 737. q g^j^d 3 centimeters wide at the inlet 

orifice A, which was well rounded off, the 

following result was obtained : 

3) - = 1,0453 + 0,0001137 h 
s 

+ 0,00007981 h% 

and, on the contrary, with the truncate^, 
mouth-piece A B, Fig. 737, whose width 
was 1,41 centimeters at the outlet B, the 
result was 

4) - = 1,0216 + 0,0007294 h 
s 

+ 0,00003036 h\ 

By the aid of these formulas the follow- 
ing table of the heights of jets has been 
calculated. 




Height due to velocity Jl = 


10 

9,61 
9,715 
9,48 
9,69 


20 

18,31 
18,69 
18,53 
19,08 


30 

25,98 
26,75 

26,77 
28,02 


40 

32,58 
33,77 
33,97 
36,39 


50 

38,12 
39,72 
39,98 
44,09 


60 

42,66 
44,63 
44,79 
51,08 


70 

46,30 

48,58 
48,47 
57,31 


Height of jet according to (1) . . 

" " " (2).. 
" " . " (3) . . 
" " " (4). 



Example. — ^If tlie pipe conducting the water to a fountain is 350 feet 
long and 2 inches in diameter, and if the conical orifice is ^ inch wide, how 
high would the jet rise under a head of 40 feet, provided all the resist- 
ances, except the friction, are small enough to be neglected ? 

Here if we put 



Ci = 0,025, Co = 0,5, ^|-y=a)< 
the height due to the velocity of efflux is 



1.)^ = .rl^r and -^ 



T3T 



350 



= 2100, 






K 



40 



(^« - ^' 4) & 



1+ (Co + Cl 



1 + (0,5 + 0,025 . 2100) . ^ 



40 
1,207 



= 33,14 feet, 



and therefore the height to which the jet will rise in still air is 



^ 435.] 



THE FLOW OF WATER THROUGH PIPES. 



881 



7i 



33,14 



1,0216 + 0,0007294 h + 0,00003036 h^ 
33,14 



1,0216 + 0,02417 + 0,03334 



1,0791 



= 30,71 feet. 



Fig. 738. 




§ 435. Piezometer. — The head, lost by the water which is 
passing through a set of pipes A B C D E, Fig. 738, in conse- 
quence of contractions in 
the conduit, friction, etc., 
can be measured by means 
of the columns of water 
maintained in the vertical 
tubes B K, CM, D which 
are attached to the pipe ; 
when they serve for this 
purpose only, they are called 
piezometers (see § 386). 
If V is the velocity of the water at the point B, Fig. 738, where 
a piezometer is inserted, I the length and d the width of the por- 
tion A B of the pipe, li the head of water or depth of the point B 
below the level of the water, ^^ the coefficient of resistance for the 
entrance of the water from the reservoir into the pipe and ^ the 
coefficient of friction, we have the height of the piezometer, which 
measures the pressure in B, 

^ = ^-(l + ^« + 4)2T . 
On the contrary, if the length of the portion B C of the pipe is 
li and the fall is Ai, we have the height of the piezometer at 

^■ = * + *'-(i + ^' + ^1 + 4)4 

Hence the difference of the heights of the piezometer is 

. -z-h -C^ ^ 

and, inversely, the height of resistance of the portion B G of the 
pipe is 

i^- . -— — hx -V z — Zx=^ fall of this portion of the pipe plus 
a Z g 

the difference of the heights of the piezometers. 

We see from this example that the piezometer can be employed 

to measure the resistances, which the water has to overcome in 

passing through the pipes. If any obstacle, if, e.g., a small body 

sticks fast in the pipe, its presence will be shown immediately by 

the sinking of the column of water in the piezometer, and the dis- 



882 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 435. 



tance it sinks will indicate the amount of this resistance. The re- 
sistances occasioned by regulating apparatuses, such as cocks, valves, 
etc. (a subject which will be treated in the following chapter), can 
also be expressed by the height of the piezometer. Thus the 
piezometer at D is lower than at C not only on account of the fric- 
tion of the water in the portion C I) of the tube, but also on ac- 
count of contraction in the pipe produced by the valve gate S. If, 
when the valve-gate is completely open, the difference JV of the 
heights of the piezometers = Jh and if, when the gate is pushed in 
a certain distance, it is p ho, the difference, or sinking, hi — ^2, 
gives the height of resistance due to the passage of the water 
through the valve gate. 

Finally, the velocity of efflux of the water can be calculated 
from the height of the piezometer.,. If the height of the piezometer 
P Q = z, the length of the last portion of the tube D E = I and 
the width of the same = d, we have 






V — 



V 



TT— and therefore the velocity of efflux is 

^9 

^ /d 2 a z 




Example. — If the height of the piezometer P Q = z upon the system 
of pipes in Fig. 738 is f feet, if the length of the pipe P £J, measured from 
the piezometer to the outlet orifice, is Z = 150 feet and if the diameter of 
the tube is 3|- inches, it follows, when the coefficient of resistance C = 0,025, 
that the velocity of efflux is 

V = 8,035 j/- 
and the discharge 



3,5 



0,75 



150 . 12 • 0,025 



= 8,025 . 0,2415 = 1,94 feet, 



jA . 1,94 = 0,129 cubic feet. 
Remark. — The motion of water in a pipe B G P, Fig. 739, can easily 
, Fig. 739. 





be disturbed by air, which may be given off from the water or enter the 
pipe from without. In order to prevent either case from occurring, we must 



436.] RESISTANCE TO THE MOTION OF WATER. ETC. 883 

take care that the pressure at every point shall be positive, or rather that 
it shall exceed the atmospheric pressure, or that there shall be a column 
of water C -E^ in everj'^ piezometer. The height of this column is 



. = A, .- (i + c. .+ ;'i) f^, 



h^ denoting the head C at CJ^the length of the portion B C of the pipe 
and V the velocity of the water in the tube. It is, therefore, necegsary that 



A, > (l + f» + C'j) f^, 



that, E.G., the head of water in the receiving reservoir shall at least exceed 
the height due to the velocity of the water>in the pipe. Otherwise the 
pipe may suck in air in an eddy. 

We can also put ft, > j- !>, h denoting the entire fall HK 

1 + :, - C^ 

of the pipe and I its entire length BCD. 

If we wish to prevent the air from accumulating in the pipe, we may 
lay the pipe in such a position that it will rise slightly in the direction in 
which the water is moving. The air will then be carried along with the 
water. 



CHAPTER IV. 



RESISTANCE TO THE MOTION OF WATER WHEN THE CONDUIT 
IS SUDDENLY ENLARGED OR CONTRACTED 

§ 436. Sudden Enlargement. — Changes in the cross-section 
of a pipe or any other conduit produce a change of velocity. The 
Telocity is inversely proportional to the cross-section of the stream ; 
the wider the vessel is, the smaller is the velocity, and the narrower 
the vessel is, the greater is the velocity of the water flowing through 
it. If the cross-section of a vessel changes suddenly, as, e.g., that 
of the tube A C U, Fig. 740, does, a sudden change of velocity, 
„ „.^ accompanied by a loss of vis viva 

and a corresponding diminution 

A ■ f -'(gz^w:::z^^-r^- " - ^ ^^ pressure, takes place. • This 

— ^j V ^ ^ " -^^-'"^'^^f ^'^v- loss is calculated in exactly the 

~"^ I^H^: ^;-^ ;-/ ^ "_: same manner as the loss of me- 

^ G chanical effect occasioned by the 

impact of inelastic bodies (see § 335). Every element of the water, 



884: GENERAL PRINCIPLES OF MECHANICS. [§436. 

which passes out of the narrower tube B D into the wider one D G^ 
impinges against the more slowly moving current in this pipe and 
after the impact moves forward with it. Exactly the same phe- 
nomena occur when solid inelastic bodies collide ; these bodies 
also move forward after the impact with a common velocity. Now 
we have found that the loss of mechanical effect occasioned by 
the impact of inelastic bodies is 

J _ (^1 — -^2)' G\ Gi 

and since in this case the impinging element G^ is infinitely small 
compared to the mass of water 6^2 impinged upon, we can put 

^ ^ (^1 - v^y ^ 

and consequently the corresponding loss of head is 

Hence, hy the sudden change of velocity, a loss of head is caused^ 
which is measured hy the height due to this change of velocity. 

Now if the cross-section of the one pipe AC, — F^, that of the 
other pipe C E, which is united to it, = F, the velocity of the water 
in the first tube = Vx and that in the other = v, we have 

Fv 

"^ = -1^ 

and therefore the loss of head in passing from one tube to the 
other is 

*■ = (J - ^)- f^ 

and the corresponding coefficient of resistance, which was first 
found by Borda, is 

The head 

^' = S - ^)- 1/ 

which we have just found, cannot of course be lost without pro- 
ducing any effect; we must rather assume that the mechanical 
effect corresponding to it is employed in separating and communi- 
cating a vibratory motion to the elements of the water, which before 
formed a continuous mass, and in forming the eddies W, W. 

The experiments made by the author confirm this theory. If 



§ 437.J RESISTANCE TO THE MOTION OF WATER, ETC. 885 



the tube D 6^ is to be maintained full of water it must not be very 
Fig 741. short or much wider than the tube 

AC. The loss is done away with 

•^^^— -^g ^ ^f^^^HP:^ ^ when, as in Fig. 741, the edges are 

"^'^^^^V^~^;^2Sc^-55^""^^^ rounded off so as to cause a gradual 

^ ^*^c'^'"^''™™''~" G passage from one tube into the other. 

Example. — If tiie diameter of one of the portions of the compound 

F 
pipe, Fig. 740, is twice that of the other, then -^ = (f)^= 4, the coefficient 

of resistance ^ = (4 — 1)^ = 9 and the corresponding height of resistance 

for tlie passage from the narrower to the wider tubes is = 9 . — . If the 

t7 

velocity of the water in the latter pipe is = 10 feet, it follows that the 
height of resistance is = 9 . 0,0155 . 10'^ = 13,95 feet. 

§ 437. Contraction. — A sudden change of velocity also takes 

place, when the water passes from a vessel A B, Fig. 742, into a 

narrower pipe D G, particularly if at the place of inlet C D there 

is a diaphragm with an opening, whose cross-section is smaller than 

the cross-section of the pipe D G. If the area of this orifice = F^ and 

if a is the coefficient of contraction, we have the cross-section of the 

contracted stream F^ = a Fx\ and if, on the contrary, F is the 

cross-section of the pipe and v the velocity of efflux, we find the 

velocity of the water at the contracted cross-section F<i, by means 

of the formula 

F 



v^ = 



aF, 



hence the loss of head in passing from F^ to F or from v^ to v is 

{v, - vY IF 



li = 



29 



- \a, F, / 2 / 
and the corresponding coefficient of resistance is 



Fig. 742. 



Fig. 743. 



>x%^v^,^^^sv-.^NNN^■^N^^^^^N 






!^N■^N^V--.■,x-c^^,x^^.^.N^^^■.^^x^^^~x- 









D E 









K: G 



B 



If the diaphragm is absent, we have a common short pipe. Fig, 
743, and then F — F^ and 



886 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 437. 



or inyersely 



< - e - ')■■ 



a = ~ 



1 + v^ 

Assuming a = 0,64, we obtain 



^ = (^-)'= (")^ = 0,316. 



^ is increased by the resistance at the entrance into the tube and 
by the friction of the water in the exterior portion of the tube to 
0,505 (§ 422). 

From experiments made with a short tube, the inlet orifice of 
which luas contracted as is represented in Fig. 742, the author has 
been led to the following conclusion : The coefficient of resistance 
for the passage of the water through the diaphragm and into the 
wider tube can be expressed by the following formula : 



^=t-^-r= 



but we must put 




















: for 5 = 


0,1 
0,616 


0,2 


0,3 


0,4 


0,5 


0,6 


0,7 
0,603 


0,8 
0,601 


0,9 



0,598 


1,0 


a = 


0,614 


0,613 


0,610 


0,607 


0,605 


0,596 



and consequently 



C = 



231,7 



50,99 



19,78 



9,612 



5,256 



3,077 



1,876 



1,169 



0,734 



0,480 



If, E.G., the narrow cross-section is half that of the pipe, the co- 
efficient of resistance is C = 5,256, I E. the passage through this 
contracted orifice occasions a loss of head 5^ times as great as the 
height due to the velocity. 

Example. — What is the discharge through the apparatus represented 
in Fig. 742, when the head is 1^ feet, the diameter of the contracted circu- 
lar orifice 1|, and that of the pipe G B, = 2 inches ? Here we have 

w /li\'^ 

^ = I y ) = (1)2 = A = 0,56 and therefore a = 0,606, and 

r ^ / 16 y _ / 16 - 5,454 y _ /10^46y _ 

^ V9. 0,606 / \ 5,454 / ~ \ 5,454 / ~ '^' * 



§438.] RESISTANCE TO THE MOTION OF WATER, ETC 887 

Now if we put A = (1 + C) g-? we obtain the velocity of efflux 

V = = — ' = 4,51, 

Vl + C V4,74 

and consequently the discharge is 

TT d^ TV 

^ ^ __. ^ := _ . 4 . 12 . 4,51 = 54,12 . tt = 170 cubic inches. 

§ 438. Influence of Imperfect Contraction. — In the case 
considered in the last paragraph, where the water comes from a 
large vessel, the contraction can be considered as perfect ; but if 
the cross-section of the vessel, or that of the stream which arrives 
at the narrow orifice, is not very great compared to the cross-sec- 
tion Fi, Fig. 744, of that orifice, the contraction is imperfect, and 
the coefficient of resistance is consequently smaller than in the case 
just considered. If the notations previously employed are retained, 
we have again the height of resistance or the head lost in passing 
through Fi 

. - (^ _ iV ^' 
^ ~ \a, F, V 2 / 

but we must substitute variable values for a, which increase with 
the ratio 7^ of the cross-section of the narrow orifice to that of the 
pipe, which conducts the water to it. If a diaphragm is placed in 

Fig. 744. Fig. 745. 

A A D ~E 






" "^i'i'"_''- ""--tT"r"-'"i^'"''^'^^-^!i^:^^j:''_'^!^X^'ii'l''-3 



^:^-li?^ 






5i!.G£==K;3i 



f 









-l-...-^...^^^ — ^..^^.-^ . 



&t^'^ n G B C G 



13 

a pipe A G, Fig. 745, of constant diameter, the same reasoning 

F, 
holds good ; but the coefficient a depends upon -^ 

According to the author's experiments, we must substitute in 
the formula for the coefficient of resistance 



' - (6^. - ')'• 



888 



GENERAL PRINCIPLES OF MECHi^JN'ICS. 



[§489. 





0,1 
0,624 


0,3 


0,3 


0,4 


0,5 


0,6 0,7 
0,713 0,755 


0,8 


0,9 


1 
1,0 i 


a^ = 


0,633 


0,643 


0,659 


0,681 


0,813 


0,893 


1,000 



whence it follows that 



C = 



335,9 



47,77 



30,83 



7,801 



3,753 



1,796 



0,797 



0,290 



0,060 



0,000 



If b^/ rounding off the edges the contraction is diminished or 

preyented, the loss of head be- 
comes smaller, and it can be done 
away with, almost entirely, by in- 
troducing into the pipe a piece, 
which widens gradually and is 
shaped like M W, Fig. 746. 
Example. — What head is necessary, if the apparatus represented in Fig. 
747 is required to deliver 8 cubic feet of water per 
minute ? Let the width of the diaphragm F^ be = 
1^ inches, the width of the discharge-pipe i> (x, = 3 
inches, and the width of the vessel J. (7, = 3 inches, 
then we have 




Fm. 747. 




now 



V = 



^ = l~^\ =1 whence a = 0,637 ; 

and the coefficient of resistance 

^ = (97o;637 ~ V "" (^37 = ^'^^^• 

Hence it follows that the velocity of efflux is 

4 Q 4.8 19,2 , ^ 

-^ = 6,112 feet, 



ird^ 60 . TT (1)3 

and, therefore, the required head is 

h = (l + n -^ = 4,307 . 0,0155 . 6,113^ = 3,43 feet. 

§ 439. Relations of Pressure in Cylindrical Pipes.— By 

Fig 748 ^^^^ ^^^ ^^ Borda's formula we 

jj can calculate the various rela- 

^ tions of the pressure in a dis- 




■¥-^^v charge pipe, the diameter of 



----^" ^' which is not constant. Let pi 
^ be the pressure and Vi the ve- 



§439.] RESISTANCE TO THE MOTION OF WATER, ETC. 889 

locity of the water at F^, and p the pressure and v the velocity of 
the same at F, then we have 

- + 5- + - \ ' =z ^ + ' and therefore 

V\ 19 , 'o' — Vx + (vi — v)' p ivx —v)v 

^^ = - H ^ = - -'^ —, or 

7 7 2g 7 g 



r r \-^i /g 



(5 - ' )f 



But the total head is 
n = ^ h 






hence we have also 

y "~ y v' + (vi — v)'' 



h. 



-P _ 



2(J-l)^. 



When a stream of water, whose cross-section is F, flows into the 
free air, ^ is — to the height I? of the water barometer, and there- 
fore the height of the piezometer at F^ is 



iSfl = -^ = ^ — 



2(^--i)a 



iWC- 



K-y 



So long as j) remains positive, the water will discharge oi E G 
with the cross-section F filled; if, on the contrary, p becomes 
negative, the supposed condition of efflux ceases to exist and the 
water flows through the exterior tube C E, as if it were not there, 

with the theoretical velocity v\ = V2 g h. 

In order to have a full discharge at E G, it is necessary that 

3 (J - 1) '' 

^ ' ' < J, or that 



1 + 

1 + 



h . 






2 



If, then, the hmits of the head h, given by this formula, are sur- 
passed, the discharge with a full cross-section ceases. 



890 GENERAL PRINCIPLES OF MECHANICS. [§ 489. 

These formulas are also applicable to the case of the pipe C E, 
Fig. 742, with a diaphragm ; but here we must substitute instead 
i^i, Oj F^ ; hence, for efflux with a filled tube, we must have 

h \«i F^ I 



' ,'" 



(A-) 



If we remove the diaphragm, we have a simple short pipe O E, 
Fig. 743, and then Fx = P; hence we must put 



7 1 + 

-< — 



t-^ 



e-) 



If we substitute a = 0,84 or - — 1 = 0,5625, we obtain the 

limit of discharge with filled cross-section through these pipes 

A 1 + 0,3164 ^i / T -, ^ 
l^ 2.0,5625 '^''^'l ^ ' '• 

If we assume J = 34 feet, it follows that when the head is 
greater than 1,17 . 34 = 39,8 feet, the efflux with a full cross- 
section through a short pipe ceases. 

The results of the author's experiments coincide perfectly with 
the above conclusions (see the article upon the efflux of water under 
great pressure m the 9th volume of the " Civilingenieur"). 

This limit is reached more quickly, when the water discharges 
into rarefied air ; for m that case b is less than 34 feet. If, e.g., the 
height of the water barometer in this space was three feet, the 
efflux with filled cross-section for a short pipe would cease when 
the head became h =z 1,17 . 3 = 3,51 feet. 

If the water flows through a pipe ACE, Fig. 749, which is 
gradually enlarged, the height of the piezometer at the inlet 
portion ^ ^ is - 
Pi _^ p Vx — v^ p 



:(0-']^=f-[(a-']«- 



y y 2<7 y 

consequently, if we put - = Z>, 

Fio. 749. ^'^ ^ ^ ^ ~Vk'FJ ^ ^* 

^ ^^^ ^^___:^ ^ - We must have, therefore, 



440.] RESISTANCE TO THE MOTION OF WATER, ETC. 



891 



when the efflux takes place with a filled cross-section. If we put 

T = 1,1*^ or - = 0,8547, we obtain the ratio of the cross- section, for 

which, under a head li = 39,8 feet, the efflux with filled cross- 
section ceases, viz. • 

F 



F, 



= V 1 + 0,8547 = 1,362. 



Fig. 750. 



§ 440. The Relations of Pressure in Conical Pipes. — The 

relations of efflux and pressure in a cylindrical pipe C E, with or 
without diaphragm, undergo the following modifications, when an- 
other mouth-piece or another tube E G H K, Fig. 750, is added to 

the former. Let F denote the 
cross-section, v the velocity and p 
the pressure of the water at the 
outlet H K, F^ the cross-section of 
the inlet, a F^ that of the con- 
tracted stream of water, v^ the ve- 
locity and fx the pressure of the 
water in the latter ; in like man- 
ner let i^2 be the cross-section of the tube, where the stream of 
water again touches the wall, v^ the velocity and p^ the pressure 
of the water at that point. Then we have 

-^ nr ^ -I ^ % and therefore 




S&JSiSE 



B 



r r 

i?l _ i?2 ^2 (Vl_ 



r 



r 



9 



_p V + v^ 



^9 

y 



^'2' ^2 (^1 — ^^2) 



^g 9 

v^ — 2 Vi V2 + v^ 



y 2g 9 y 

or, since we can put a F^ i\ — F^v^ — Fv, or 



^9 



Fv , Fv 

V, — — ^ and ^2 = -pTy 

% F' 



aF, 



[^- 



+ 



©' 



^9 



y y L d F^ x^) 

Now the head necessary to produce the required velocity of 
efflux is 



V 



^9 



—^g -V^XaF, fJ ]2g' 



from which it follows that 



892 



^:=l + 



GENERAL PRINCIPLES OF MECHANICS. 



[§440. 



aF,F, 



+ 



©■ 



1 + 



/ F _ F_\^ 
WF, ~ fJ 



h 



r 






2 



+ 



d Fx F^ F^ 



P 
I.E. Zx^= ~ 

7 



F'^ \a Fx FJ 
a Fx F, \F, ^ F,y 
F' ^ \aFx Fj 



h 



V h 



or, when the water is discharged into free air, 

2 /J^ 1\ 

\Fx "^ F,^l 



Zx = h — 



aFxF, 



h. 



F, '^ \a~Fx W 
If the efflux takes place with full cross-section, we must have, 
according to what precedes, 

h ^ F' '^ [aFx fJ 



b^ 



2 _ (1^ IV 
aFxF, \F'^ F,'l 



ill 



^ \a Fx F, F:J h \aFx Fj' 



By the aid of the foregoing formula the relations of the efflux 
through the conical tubes A B D E, Figs. 751 and 752, can be 



Fig. 751. 



Fig. 752. 



*> - 





gsKSV.~^-»^0 



given by substituting for F^ the cross-section of the pipe, where the 
stream touches the wall. If 6 denotes the semi-angle of diver- 
gence A C B of one, or the semi-angle of convergence of the other 
tube, and if we assume that the length Fx F<, of the eddy is equal to 
the width ^ ^ = ^ of the orifice, we have the width of pipe, where 
the stream reaches its wall, 

t?2 = ^^ -fc 2 dx tang. ^ = (1 ± 2 tang. 6) dx, 



§440.] RESISTANCE TO THE MOTION OF WATER, ETC. 893 

and therefore the ratio of the cross-sections 

Fa I dcf 



3 = (|l) = (1 ± 2 tang.S); 



m which the positive sign is to be employed for the diyergent pipe 
in Fig. 751 and the negative sign for the convergent one in Fig. 
752, E.G. for S = 2A degrees, 2 tang. 6 = 0,0875 and 

^ = (1 ± 0,0875)^ either = 1,1827 or 0,8327 ; 

hence the velocity of efflux in the first case is 



% g h _■/ ^ g h 



"V-(^-™)'©'"l/i-^"(t' 



and, on the contrary, in the second 






The corresponding coefficient of efflux 

1 



4/1 + 0,514 (jy 



for the divergent tube is, of course, considerably smaller than the 

coefficient of efflux 

1 



\/l + 0,1308 (^ 



of the convergent tube. 

If, E.G., the tubes were three times as long as wide at the inlet 
orifice, we would have in the first case 

(^y=: (1 + 6 tang. 6)* = 1,2625^ = 2,5405 and 

ft = — = 0,659, and, on the contrary, in the second case 

V 2,306 

{^)'= (1-6 tang. 6)* = 0,7375* = 0,2958 and 
■^ = 0,981 (compare § 425). 



V 1,0387 

If the efflux through these pipes takes place with filled cross- 
Bection, we must have 



894 



GENERAL PRINCIPLES OF MECHANICS. 



[§44L 



J. 1 + 

h ^ 2F_F 
a F, F, 
or in the first case, when 

F 1,5939 



/ F __ ^Y 

\ a F, Fj 



-['-©■]• 



aF, 



0,64 = ^'^^''' ^^^ F. 
1 + 1,1429' 



F 1,5939 



1,1827 
2,3062 



= 1,3477, 



I ^ 6,7112 - 2,8163 3,8949 



= 0,592, 



and the head h must be less than 34 . 0,592 = 20,1 feet. 

§ 441. Elbo"ws. — A particular kind of impediment is opposed 
to the motion of water in pipes, when the latter are hent or form 
elboivs. These resistances cannot be determined with safety by 
theory and must, therefore, like so many of the phenomena of 
efflux, be studied by experiment. If a pipe A C B, Eig. 753, forms 
an elbow, the stream separates itself from the inner surface of the 
second branch of the pipe m consequence of the centrifugal force ; 
when this piece is short, the efflux with full cross-section ceases, 
and the discharge is, therefore, smaller than from an equally long 
straight pipe. If the exterior portion C B of the elbow A C B, 

Fia. 753. Fig. 754 





Fig. 754, is longer, an eddy S is formed beyond C, and, when the 
tube is again filled, the velocity of efflux v is smaller This dimi- 
nution of the velocity of efflux must be treated exactly in the 
same manner as the resistance produced by a contraction in the 
pipe. If F is the cross-section of the tube and F, that of the con- 
tracted vein, we have the coefficient of contraction of the latter 

F 

F,' 
and, therefore, the corresponding coefficient of resistance ' ^ 



§ 441.] RESISTANCE TO THE MOTION OF WATER, ETC. 



895 



^ = (S - ■)'- il - -I 



The coefficient of contraction a, and consequently the corre- 
sponding coefficient of resistance ^, depends upon the semi-angle of 
deviation 6^AGD = BCE^)^BGF, Fig. 753, and accord- 
ing to the experiments of the author, made with a tube 3 centi- 
meters in diameter, we can put 

C == 0,9457 sin^ ^ + 2,047 sin.' cJ. 

The following table contains a series of coefficients of resistance, 
calculated for different angles of deviation. 



(5°= 


10 


20 


30 


40 


45 


50 


55 


60 

1,861 


65 


70 


C = 


0,046 


0,139 


0,364 


0,740 


0,984 


1,260 


1,556 


2,158 


2,431 



We see from this table that the yis yiya of water in pipes 
is considerably diminished by the elbows. If, e.g., the elbow 
makes a right angle or c5 — 45°, we have the loss of head occa- 
sioned by it 

or nearly as much as the height due to the velocity. 

When the pipes are narrower, C becomes considerably greater, 
E.G., for an elbow 1 centimeter in diameter and with an angle of 
deviation of 90°, ^was found = 1,536. See the author's '^ Exjperi- 
mentalhydraulilcr 

If to one elbow A C B, Fig. 755, another elbow is joined, as is 
shown in Fig. 756, and Fig. 757, a peculiar, but at the same time 



Fig. 755. 



Fig. 756. 



Fig. 757. 






m ::|; I-; 



easily explicable, phenomenon of efflux is observed. The second 
elbow B D E, Fig. 756, which turns the stream to the same side 
as the first one ^ C^, produces no further contraction of the 



896 



GENERAL PRINCIPLES OF MECHANICS. 



[§442. 



stream, and, therefore, for efflux with full cross-section ^ is no 
greater than for a simple elbow A C B. But if the elbow B D E, 
Fig. 757, turns the stream to the opposite side, the contraction is 
a double one, and the coefficient of resistance is consequently twice 
as great as for a single elbow. If, finally, B D E \^ so joined to 
A C B that D E stands at right-angles to the plane A B B, ^ then 
becomes about 1 A times as great as for the single elbow A C B. 

Example. — If a system of pipes K L N^ Fig. 758, 150 feet long and 5 

inches in diameter, which should 
discharge 25 cubic feet of water, 
contains two elbows, the required 
head will be 



Fig. 758. 




h = (1,505 + 8,712 + 2 . 0,984) ^ 

= 12,185 . 0,1448 = 1,76 feet. 
(Compare Example in § 430.) 

§ 442. "^0X1^%.— Curved pipes, when the other circumstances 
are the same, cause much less resistance than elbows. They also 
cause, in consequence of the centrifugal force of the water, a par- 
tial contraction of the stream ABB, Fig. 759, so that, when the 
bend is not terminated by a long straight pipe, the cross-section 
El of the stream at its outlet is smaller than that E of the pipe. 
But if the bend ABB, Fig. 760, is terminated by a long straight 

Fig. 759. Fig. 760. 

A 





pipe B E, an eddy E is formed and an efflux with filled cross-sec- 
tion again takes place at the expense of the vis viva of water. If 

the coefficient of contraction -r=^ = a, we have for the coefficient 

E 

of resistance of the bend. 



^= {.'-)■■ 



a 



The coefficient of contraction a depends upon the ratio - of the 



r 



radius B M — E M = a, Fig. 759, of the pipe to its radius of cur- 



§442.] RESISTANCE TO THE MOTION OF WATER, ETC. 897 

vature C M = r, and it can be determined approximatively in the 
following manner. If v is the velocity of the water upon entering 
the bend and v^ that of the contracted vein, we have v^ F-^ ~ y ir 

F 

whence Vi = -— v, and, therefore, the head which measures the 

pressure in ^ ^ is 

'' = -Y7~ = V[F]-^¥g' 
This height, multiplied by 1 and y, gives the pressure of the stream 
of water in all directions upon the unit of surface at E 

^-'"-[©'^']f> = [©'-']r,- 

Since the centrifugal force of the water acts upon the convex 
side in opposition to the pressure p, it is possible that it may bal- 
ance the latter completely. But in this case the exterior air would 
enter and separate the stream entirely from the convex side, as is 
shown in Figs. 759 and 760. The centrifugal force of a prism of 
water, whose length \s B F = 2 a and whose cross-section is 1, is, 
when the radius of curvature is M = r, 



v' 



q = — . 2 a y. 
9 r ' 

Now if we put p = q, we have the condition of separation of the 

stream from the wall of the pipe 

J_ _ 4a 

a' T' 

and consequently the coefficient of contraction 

^ r -h 4:a^ 
hence the coefficient of resistatice for efflux with a full pipe is 



{/ 



r + 4a 



')•• 



r 

As this calculation is based upon a mean velocity and a mean 
radius of curvature, it will, of course, give but an approximate 
value of a and C 

From his own experiments and from the results of some obser- 
vations made by Du Buat, the author has deduced the following 
empirical formulas for the coefficients of resistance of water in 
passing through bent pipes : 

1) for bends with circular cross-sections 

^ = 0,131 + 1,847 [^y ; 
57 



898 



GENERAL PRINCIPLES OP MECHANICS. 



[§443, 



2) for bends with rectangular cross-sections 

<= 0,124 + 3,104(01 

The following tables are calculated according to these formulas : 

TABLE L 

Coefficients of the resistance due to the curvature of ^pes with circular cross- 
sections. 



a 

r ~~ 


0,1 


0,2 


0,3 


0,4 


0,5 


0,6 


0,7 


0,8 
0,977 


0,9 


1,0 


C = 


0,131 


0,138 


0,158 


0,206 


0,294 


0,440 


0,661 


1,408 


1,978 



TABLE II. 

Coefficients of the resistance due to the curnature of pipes with rectangular 









cross-sections. 












a 
r 


0,1 


0,2 


0,3 


0,4 


0,5 


0,6 


0,7 


0,8 


0,9 


1,0 


C = 


0,124 


0,135 


0,180 


0,250 


0,398 


0,643 


1,015 


1,546 


2,271 


3,228 



From the above tables we see that for a circular pipe, whose 
radius of curvature is twice the radius of its cross-section, the coef- 
ficient of resistance —- 0,294, and that for a pipe, whose radius of 
curvature is at least 10 times the radius of the cross-section, the 
coefficient = 0,131. 

In order to check the contraction of the stream of water in a 
bend^ B D, Fig. 761, the cross-section of the pipe must be grad- 
ually diminished in such a manner that the ratio of the cross-sec- 
tion D H ^ F,oi the outlet orifice to that ^ S' := i^of the inlet 
1 



shall be a = 



Fig. 763. 






§ 443.] RESISTANCE TO THE MOTION OF WATER, ETC. 



899 



If one bend B D, Fig. 762, is terminated by another, which 
turns the stream farther in the same direction, if, e.g., the axis of 
the pipe forms a semicircle, Hke B D E, Fig. 763, the contraction 
is not changed and a and ^ have the same values as for the pipe in 
Fig. 762, which forms but a quadrant. If, on the contrary, a bend 
D E, Fig. 764, which turns the stream in the opposite direction, is 
attached to the first one, an eddy i^is formed between the two and 
a second contraction of the stream takes place, by which the resist- 
ance (s) is nearly doubled. 

Fig. 764. Fm. 765. Fig. 766. 

A . A A 





The resistance to water flowing through bends can be dimin- 
ished by enlarging the cross-section of the pipe, as in B D E, Fig. 
765, or by inserting in it a thin ])artition, like S in B I) E, Fig. 
766 ; for in the first case the velocity v, and in the second the ratio 

- is smaller, and consequently the coefficient of resistance C is ren- 
dered less. 

Example. — If the system of pipes B L M^ Fig. 767, in the second ex- 
ample of § 430, contains 5 bends 
Fig. 767, each of 90°, and if the radius of 

curvature of each is 2 inches, we 
have 

^ ~" ^' 

and according to the first of the 
foregoing tables, the correspond-, 
ing coefficient of resistance C. =; 
0,294 ; consequently for the 5; 

bends 5 C = 1,47 ; hence the velocity of the water issuing from the pipe; 

instead of — 

17,945 




V = ,,J_ .__,.. = 6,52 feet, is 
V7,582 

17,945 



V = 



17,945 



V 7,582 + 1,47 V9,052 



= 5,964 feet, 



900 



GENERAL PRINCIPLES OF MECHANICS. 



[§443. 



so tliat the discharge per second is now 

q = 0,7854 . 3^ • 5,964 = 0,1301 cubic feet = 224,81 cubic inches. ] 

§ 443. Valve-Grates, Cocks, Valves. — In order to regulate 
the discharge of water from pipes and vessels, we employ various 
kinds of apparatus, such as cocks, valve-gates, and valves^ by means 
of which we produce a contraction in the pipe, which occasions a 
resistance to the passage of the water, the value of which is deter- 
mined in the same manner as the losses of head in the foregoing 
paragraph. As the stream of water is subjected to changes of 
direction, is divided, etc., the coefficients a and ^ can only be 
determined by experiments made for that purpose. Such experi- 
ments have been made by the author,* the principal results of 
which are given in the following tables : 

TABLE I. 

The coefficients of resistance for the passage of water through valve-gates 
or slide valves (Fr. tiroirs; Ger. Schieber or Schubventile) mpa/ralleh- 
pipedical pipes. 



Ratio of the cross 

sections -^ = 
-t 


1,0 
0,00 


0,9 


0,8 


0,7 


0,6 


0,5 


0,4 


0,3 


0,2 


0,1 
193 


Coefficient of re- 
sistance C = 


0,09 


0,39 


0,95 


2,08 


4,02 


8,12 


17,8 


44,5 








ta: 


BLE 


II. 













The coefficients of resistance for the passage of water through valve-gatetf 

or slide-valves in cylindrical pipes. 



Relative height of opening 

8 = 



1,000 


1 

"S" 

0,948 


* 


3 

■5 


4 


5 

' 8 


6 
"S" 


7 
0,159 


Ratio of the cross-sections = 


0,856 


0,740 
0,81 


0,609 


0,466 


0,315 


Coefficient of resistance C = 


0,00 


0,07 


0,26 


2,06 


5,52 


17,0 


97,8 



* Experiments upon the efflux of water through valve-gates, cocks, clacks, 
and valves, made and calculated by Julius Weisbach, or under the title " Un- 
tersuchungen im Gebiete der Mechanik und Hydraulik, etc.," Leipzig, 1842. 



§443.] RESISTANCE TO THE MOTION OF WATER, ETC. 901 

TABLE m. 

The coefficients of resistance for the passage of water through a coch (Fr. 
robinet ; Ger. Hahn) in parallelopipedical pipes. 



Angle that the cock is 
turned 6 — 


5° 

0,926 

0,05 


10° 

0,849 

0,31 


15° 
0,769 

0,88 


20° 
0,68T 
1,84 


25° 
0,604 
3,45 


30° 
0,520 
6,15 


35° 
0,436 
11,2 


40° 
0,352 
20,7 


45° 
0,269 
41,0 


50° 
0,188 
95,3 


55° 
0,110 
275 


66| 



00 


Ratio of the cross-sec- 
tions = 


Coefl&cient of resist- 
ance = 



TABLE IV. 

The coefficients of resistance for the passage of water through a cock in a 

cylindrical pipe. 



Angle that the cock is turned 6 = 


5° 


10° 


15° 

0,772 

0,75 


20° 


25° 

0,613 
3,10 


30° 
0,535 


35° 


Ratio of the cross-sections = 


0,926 


0,850 


0,692 
1,56 


0,458 


Coefficient of resistance = 


0,05 


0,29 


5,47 


9,68 


Angle that the cock is turned 6 — 


40° 


45° 


50° 


55° 


60° 


65° 


82i° 



Ratio of the cross-sections = 


0,385 


0,315 


0,250 


0,190 


0,137 
206 


0,091 
486 


Coefficient of resistance = 


17,3 


31,2 


52,6 


106 


00 



TABLE V. 

The coefficients of resistance for the passage of water through throttle-valves 
(Fr. valves; Ger. Drehklappen or Drosselventile) in pa/rallelopipedical 
pipes. 



Angle that the valve is turned 6 = 


5° 


10° 


15° 


20° 

0,658 
1,34 


25° 
0,577 


30° 

0,500 
3,54 


35° 


Ratio of the cross-sections = 


0,913 

0,28 


0,826 


0,741 


0,426 

5,7 

1 


Coefficients of resistance = 


0,45 


0,77 


2,16 



902 



GENERAL PRINCIPLES OF MECHANICS. 



[§444. 



Angle that the valve is 
turned 6 = 


40° 


45° 


50° 


55° 


60° 


65° 


70° 


90° 


Ratio of the cross-sec- 
tions = 


0,357 


0,293 
15,07 


0,234 


0,181 


0,134 


0,094 


0,060 





Coefficients of resistance = 


9,27 


24,9 


42,7 


77,4 


158 


368 


00 



TABLE YI 

Coefficients of resistance for the passage of water through ihrottle-Dalves in 

cylindrical pipes. 



Angle that the valve is turned 6-. 



Ratio of the cross-sections = 



Coefficient of resistance 



5° 



0,913 
0,24 



10° 



0,826 



0,52 



15° 



20° 25° 



0,741 [0,658 



0,90 



1,54 



0,577 



2,51 



30° 



0,500 



35° 



0,426 



3,91 



6,22 



Angle that the valve is 
turned 6 = 

Ratio of the cross-sec- 
tions = 



Coefficient of resistance = 



40° 

0,357 

10,8 


45° 

0,293 

18,7 


50° 


55° 

0,181 


60° 


65° 



0,094 


70° 

0,060 
751 


0,234 


0,134 


32,6 


58,8 


118 


256 



90° 



GO 



§ 444. With the aid of the coefficients of resistance, given in 
the above tables, we can find not only the loss of head for a certain 
position of the valve-gate, cock or valve, bnt also the position we 
must give to these apparatus in order to produce a certain velocity 
of efflux or a certain resistance. Of course, such a determination 
will be more accurate, the more the regulating apparatus resembles 
that used in the experiments. Besides, the values given in the 
above tables are not correct, when the water, after passing the con- 
tracted orifice produced by the apparatus, does not fill the pipe 
again: In order that the efflux with a filled cross-section shall take 
place, it is necessary, when the contraction is great, that the pipe 
shall have a certain length. The cross-section of the parallelopiped- 
ical pipe was 5 centimeters wide and 2A centimeters high, and the 
diameter of the cylindrical pipe was 4 centimeters. With the slide- 



§444.] RESISTANCE TO THE MOTION OF WATER, ETC. 



903 



valve or valve-gate, Fig. 768, the cross-section is merely narrowed, 
and its shape in one pipe is a simple rectangle Fi, Fig. 769, and in 



Fig. 768. 
K 



Fig. 769. 



D 



ill: 



:\V-Xi^:I^ 



B 



„„„,,,,,„,-,, ,,,,„,^ 





the other a crescent F^, Fig. 770. "When cocks are employed, as in 
Fig. 771, there are two contractions and two changes of direction, 
and the resistance is therefore in this case very great. The cross- 



FiG. 771. 
H 



Fig. 772. 








A 








\) 






. -NX-^-^-V 


-SSV<SX^„NJ 


b- 




^ 


Wi 












S-.N-,-^-v^NNW.VN 


s-.-.xsxN-,-.v,...a 



B 



B 



sections of the maximum contractions have very peculiar forms. 
The stream is divided by the throttle-valve (or disc and pivot valve), 
Fig. 772, into two parts, each of which passes through a contracted 
orifice. The cross-sections of the contracted openings are rec- 
tangular in parallelopipedical pipes and crescent-shaped in cylin- 
drical ones. The following examples will sufficiently explain the 
use of the foregoing tables. 

Example — 1) If in a system of cylindrical pipes 3 inches in diameter 
and 500 feet long a valve-gate is introduced, and if it is raised f of the 
entire height, so as to close | of the diameter, what will be the discharge 
through it under a head of 4 feet ? According to what precedes we can 
put the coefficient of resistance for the entrance of the water into the pipe 
(To = 0.505 and the coefficient Ci of resistance of the pipe according to 
Table II, § 443, = 5,52, whence it follows that the velocity of efflux is 

8,025 V4 8,025 . 2 16,05 



D =: 



/ 



1,505 4- 5,52 + C 



I V7,025 + 500'. 4 C V7,025 + 2000 i 
d 



If we put the coefficient of friction C = 0,025, we obtain 

16,05 



V = 



V57,025 



= 2,125 feet. 



904: GENERAL PRINCIPLES OF MECHANICS. [§ 445. 

But the velocity 10 = 2,125 feet corresponds more accurately to C = 0,0265, 
hence we have more correctly 

« = 4^ = 2,07 feet, 
V60,025 
and the discharge per second is 

^ = ^ . 9 . 12 . 2,07 = 55,89 tt = 176 cubic inches. 

2) A system of pipes 4 inches in diameter discharges under a head of 
5 feet 10 cubic feet of water per minute ; at what angle must a throttle 
valve, placed in them, be turned to cause a discharge of 8 cubic feet per 
minute ? The initial velocity is 

and that after turning the valve 

= ^8^ • 1^91 = 1,528 feet. 
The coefficient of efflux in the first case is 

. -^- = --i^ = 0.107; 

sl'lgJi 8,025 V5 

hence the coefficient of resistance is 

= 7 ~ ^ ^ MOT"^ ~ ^ = ^^'^^' 

and the coefficient of efflux in the second case is 

= T-^o . 0,107 = 0,0856 ; 
hence the coefficient of resistance is 

= 0;08T6^ ~ ^ ^ ^^^'^' 
and the coefficient of resistance of the throttle valve 

C= 135,5 — 86,34 = 49,16. 
Now Table VI, § 443, gives for the angle 6 = 50°, f = 82,6 and for the 
angle (5 = 55°, C = 58,8 ; we can, therefore, assume that, when the valve 

16,56 

is placed at an angle of 50° + ^o o • ^° — ^3° 10', the required quantity 

of water will be discharged. If we take into consideration the fact that 

the coefficient of friction changes from 0,0268 to 0,0283 when the velocity 

decreases from 1,91 to 1,528, we have more correctly 

283 
C = 135,5 - 86,34 ^r~ = 135,5 - 91,2 = 44,3, 

268 

and consequently the angle that the valve must be turned 

*5 = 50° + ^ 5° = 52° 14'. 
2o,2 

§ 445. Valves. — The knowledge of the resistance produced by 
valves (Fr. soupapes ; Ger. Ventile) is of the greatest importance. 
Experiments have also been made by the author with them. 
Those most commonly employed are the puppet valve and the 



§ 445.] RESISTANCE TO THE MOTION OF WATER, ETC 



905 



clack valve, which are represented in Figs. 773 and 774. In both 
cases the water passes through an aperture in a ring R G, which 

Fig. 773. Fig. 774. 

^ E, _I) 







is called the seat. The puppet valve K L, Fig. 773, is provided 
with a spindle, which runs in guides and which permits the valve 
to move only in the direction of its axis ; the clack K L, Fig. 774, 
on the contrary, opens by turning like a door. We see that in 
both apparatuses not only the ring, but also the valve are obstacles 
to the motion of the water. 

The ratio of the aperture in the seat of the puppet valve, with 
which the experiments were made, to that of the pipe was 0,356, 
and, on the contrary, the ratio of ring-shaped surface around the 
open valve to the cross-section of the pipe was = 0,406, hence we 

Fi 
can put as a mean -^ = 0,381. By observing the efflux for differ- 
ent positions of the valve it was found that the coefficient of resist- 
ance decreased with the lift of the valve, but that this decrease was 
very inconsiderable, when the lift exceeded one-half the width of 
the orifice. Its value for this position was = 11, and the height 
of resistance or loss of head was 



z = ^ 



V denoting the velocity of the water in the full pipe. This num- 
ber can be used to find the coefficients of resistance corresponding 
to other relations of cross-section. If we put in general 

F^ 

KaF, 

we obtain for the case observed 



< = (A - ■)•. 



F 

and therefore 



= «'^81and?=( 1 -iy=ll, 



a = 



0,608, 



0,381(1+ Vn) 4,317.0,381 
/tnd finally the general expression for the coefficient of resistance 



906 GENERAL PRINCIPLES OF MECHANICS, [§445. 

If, E.G., the cross-section of the aperture is one half that of the 
pipe, the coefficient of resistance becomes 

= (1,645 . 2 - 1)^ =: 2,29^ = 5,24. 
In the experiments with clack-valves the ratio of the cross- 

section of the aperture to that of the pipe, i.e., -=, was = 0,535. 

The following table shows how the coefficients of resistance de- 
crease as the opening increases. 

TABLE OF THE COEFFICENTS OF RESISTANCE FOR 

CLACK-VALVES. 



Angle of opening 


15" 
90 


20" 
62 


25" 
42 


30" 
30 


35^ 
20 


40" 
14 


45" 
9,5 


50" 

6,6 


55" 
4,6 


60" 
3,2 


65" 
2,3 


70" 
1,.7 


Coefficient of resistance.. 



By the aid of this table the coefficient of resistance for clack- 
valves can be calculated approximatively, when the relations of the 
cross-sections are different. We must adopt the same method as 
we did for puppet valves. 

Example. — A force-pump delivers every time the plunger descends in 
4 seconds 5 cubic feet of water, the diameter of the column pipe in which 
the puppet-valve is placed is 6 inches, the interior diameter of the valve- 
ring is 3|- inches, and the maximum diameter of the valve is 4-1- inches ; 
what resistance is to be overcome by the water in passing through this 
valve ? The ratio of the cross-sections for these apertures is 

QiJ = (t'^>' = ^'^^' 

and the ratio of the ring-shaped contraction to the cross-section of the 
tube is = 1 - (^y = 1 - (1)^ = 0,44; 

hence the mean ratio of the cross-sections is 



F^ _ 0,34 + 0,44 



= 0,39, 



F 2 

and the corresponding coefficient of resistance 

The velocity of the water is 



V = 



4 - riv 



20 
= — = 6,37 feet, 



§446.] RESISTANCE TO THE MOTION OF WATER, ETC, 



907 



the liciglit due to the velocity is = 0,630 feet, and consequently the height 
of resistance is = 10,4 . 0,630 = 6,55 feet. The amount of water raised in 
a second weighs f . 63,5 = 78,125 lbs. ; the mechanical effect consumed 
by the passage of the water through the valve in that time is therefore 

= 6,55 . 78,125 = 511,72 foot-pounds. 



§ 446. Compound Vessels. — The foregoing theory of the re- 
sistance due to the passag'e of water through contractions, is also 
applicable to the discharge from compound vessels. The apparatus 
A B, represented in Fig. 775, is divided by two walls, which contain 
the orifices 7^ and jF], into three communicating 
vessels. If the dividing walls were absent and 
the edges at the passage from one vessel to the 
other rounded off', we would have, as in the case 
of a simple yessel, the velocity of efflux 




V = 



V2g h 



in which h denotes the depth of the orifice below 
the level of the water and ^^ the coefficient of re- 
sistance for the passage of the water through the orifice K 

But since when the water has passed through the orifices F^ and 
F.2 the cross-sections a F^ and a F^ change suddenly into the cross- 
sections 6^1 and G<i of the vessels C D and B C, and according to 
§ 437 the resistances thus produced are 






\aF, ) \ G J 2g~ \F, G )' 2g 



9 



and 



^'2g~\aF, / \ G, / 2 g ~ \F, G, I ' 2 g' 



we have 



^^''^^^¥^^^^2-a''^^YcrV-'^^AFr-G)^ 



(J '^g 'ig 
whence we obtain the velocity of efflux 



V — 



V2gh 



.f 1 TF aFV IF aFV 



908 



GENERAL PRINCIPLES OF MECHANICS. 



[§446. 



In the compound vessel, represented in Fig. 776, from which 
the water is discharging, the same conditions exist, but perhaps 

we must here consider the fric- 
FiG. 776. iIq-j^ of i]^Q water in the com- 

municating tube C E. Let I be 
the length, di\\Q diameter, ^the 
coefficient of friction of this 
tube, and v^ the velocity of the 
water in it, then we have for 
the head lost by the water in 




*. = [! + G 

or, since the velocity Vi — 



- 1 
a F 



passing from A C to G L 
l_ 






V, 



Ai = [ 



1 4- 



£->)■-<' 



d- 



(t)' 



^9 



If we subtract this height from the total head h, there remains the 
head in the second vessel h^ =^ h ~ hi', hence the velocity of 
efflux is 

V2y^ _ '♦^27^ 



V 



This determination becomes very simple when the apparatus is 

like the one represented in Fig. 777 ; 
for in this case we can assume the 
cross-sections G, Gi, G^ to be infi- 
nitely large, compared to the cross- 
sections of the orifices F, F^, F^, 
The first difference of level H, or 
the height of resistance for the pas- 
sage through i^i, IS 

""~ 2g\aJ~ W, fJ ' 2 / 
and in like manner the second difference of level Oi Hi or the 
height of resistance for the passage through F^ is 

^'- \a,Fj '2g' 
in which a, ai and a^ denote the coefficients of contraction for the 
orifices F, Fi and F^. Hence 




446.] RESISTANCE TO THE MOTION OF WATER, ETC. 909 



V2gh 



and the discharge is 



. aF V2gh 



_ V'Zg/i 

It is easy to see that under the same circumstances compound 
yessels, or reservoirs, discharge less water than simple ones. 

Example.— If in the apparatus, Fig. 776, the total head or the deptli 
of the centre of the orifice F below the level of the water in the first vessel 
is = 6 feet, if the orifice is 8 inches wide and 4 inches high and if the 
reservoirs are united by a pipe 10 feet long, 13 inches wide and 6 inches 
high, what will be the discharge ? 

The mean width of the trunk is 

^ "^ " 2.15' "^ ^ ^^^^' ^®°^® d ^ ~T~ "^ ^^' 
Putting the coefficient of friction C = 0,025, we obtain 

I 
^-=z 0,025. 15 = 0,375, 

and adding Co = 0,505, the coefficient for the entrance into prismatical 
pipes, we have 

1 + /I _ ij + C^ = 1 + 0,505 + 0,375 = 1,88. 

a F 64 . 8 . 4 
Since -^i^r- = -^^-tt^ — ;; — = 0,2845, it follows that the coefficient of resist- 
F^ 12 . o 

ance for the entire pipe is = 1,88 . 0,2845^^ = 0,152, and if we put the 

coefficient of resistance for the passage through F, = 0,07, we obtain the 

velocity of efflux 

Vl,07 + 0,152 V 1,222 

The contracted cross-section is 0,64 . 1 . ^ = 0,32 square feet, and there- 
fore the discharge is 

Q sr 0,33 . 17,78 = 5,69 cubic feet. 



dxO 



GENERAL PRINCIPLES OF MECHANICS. 



[§44T. 



CHAPTER Y. 



OF THE EFFLUX OF WATER UNDER VARIABLE PRESSURE. 

§ 447. Prismatic Vessels. — If a vessel, from which water is 
issuing through an orifice in the side or bottom, receives no sup- 
ply of water from any other source, the level of the water will 
gradually sink, and the vessel will finally become empty. Now if 
the discharge Q into the vessel is greater or less than that 

fi F V2 g U from it, the water level will rise or sink, until the head 

becomes h — ^r— (--^) , and afterwards the head and velocity of 

efflux will remam constant. Our problem now is to determme the 
dependence upon each other of the time, of the rising or sinhing of 
the surface of the loater and, if it occurs, of the emptying of th& 
vessel, tuhen the latter has a given form and size. The most simple 
case is that of efflux through an orifice in the bottom of a prismatic 
vessel, which receives no supply of water. Let x be the variable 
head F P, F the area of the orifice and G the cross-section of the 
vessel A C, Fig. 778, then the theoretical velocity of efflux is 



Fig. 778. 




V =^ V2 g X, 
and the theoretical velocity of the sinking surface 
of the water is 



F 



F 



V 



G G 

and the effective velocity 

II F 
~G' 



V2 



gx, 



V, 



V2g X. 



In the beginning x = F = h, and at the end of the efflux 
■a; = 0, the initial velocity is therefore 

fjiF 



G 
and the final velocity 

" Ci = 0. 

We see from the formula 



V27^, 



^.=i/^(^y... 



§448.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 911 
that the motion of the surface of the water is uniformly retarded, 

7^ ) ^ ; we also know (§ 14) 
that this velocity will be = and that the efi9.ux will cease after a 



time. 



I.E. 



'^ GVh 

v ■ — 



^* ' m' - ^v^'. 



We can also put 
t = 



2 Gh 2 Gh 2V 



^^V2jh Q Q' 

and consequently we can assume that a volume V = G h oi water 
will be discharged through an opening F in the bottom under a 
head decreasing from h to in double the time that it would, if the 
head were constant and equal to h. 

As the coefficient of efflux ii is not perfectly constant, but in- 
creases when the head diminishes, we must employ a mean value 
of this coefficient in our calculations. 

Example. — In what time will a parallelopipedical box, whose cross- 
section is 14 square feet, empty itself through an orifice in the bottom, 
which is circular and 3 inches in diameter, when the initial head is 4 feet ? 
Theoretically the time required would be 

^ a-l^V;^ = ^---1^^^ = ^^ = 319",9 = 5 n>in. 19,9 sec. 
8,025. ^.gy '•"'^'^ «'»''^ 

At the end of half the duration of the efflux the head is = {\y h = 

i . 4 = 1 foot, but the coefficient of efflux for an orifice in a thin plate, 

corresponding to a head = 1 foot, is /z = 0,613 ; hence the real duration 

of the efflux is 

319",9 



0,613 



= 521", 8 = 8 minutes 41,8 seconds. 



§ 448. Communicating Vessels. — Since for an initial head 
hx the duration of efflux is 

^ _ 2 GVh, 

' ~ iiF V2~g 
and for an initial head As the duration is 

_ 2 G\% 
[iF. V2g 



912 



GENERAL PRINCIPLES OF MECHANICS. 



[§448. 



it follows that by subtraction we obtain the time during which the 
head changes from hi to li^, or the level of the water sinks a dis- 
tance hi — ^2 ; its value is 

or, when the dimensions are expressed in feet, 

t = 0,249 ^{^Tix - '^^^). 

On the contrary, when the duration of the efl9.ux is given, we 

determine the distance s — h^ — h^ that the surface of the water 

sinks by means of the formula 

.. . ,.7^ w \^ 
ho 



II 



¥g.Ft( _. \iV^ 



%G ^)'^^ 



G 



{Vh 



4 G 



Ft 



)■ 



Fig. 779. 




The same formulas are applicable to the case of a vessel C D, 
Fig. 779, filled from another vessel A B, in which the level of the 

water is constant. If the cross-section 
of the communicating pipe or orifice 
= F, that of the vessel to be filled = G 
and initial difference of level Ox of 
the two surfaces of water = h, we have, 
since in this case the level of the water 
in the second vessel rises with a uni- 
^ i> formly retarded motion, the time re- 

quired to fill it or the time in which the second surface of the 
water rises to the level H R oi the first 
_ 2 GVh 

~ \i F . VTg 
and in like manner the time in which the distance Oi — hx be- 
tween the surfaces of the water becomes Oc^ =^ h^, or during 
which the level of the water rises a distance Oi O2 = s = hi — h^, 

2G r- r- 

^ j^ ( 4^1 - Vh,). 

\i F , \% g^ ' 

Example 1) How much does the surface of the water in the last exam- 
ple (§ 447) sink in 2 minutes ? Here we have 

A, =4, « = 2. 60 = 120,|- = j^-^, 

and if we assume also (i = 0,605, it follows that 

h - ( /j: /o- ^^Y (^ 0,605 . 8,025 . tt . 120\ 2 



t = 



§449.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 913 



= ^2 - 0,605 . 8,025 • ^^)'= 1,546^ = 2,3901 feet, 

and that the required distance that it sinks is 
s = 4 — 2,3901 = 1,6099 feet. 
2) What time does the water require to rise in a pipe CD, Fig. 780, 18 

inches in diameter, so as to overflow, when the 
pipe communicates with a vessel A Bby means 
of a short pipe 1^ inches in diameter, and when 
the surface of the water G is in the beginning 
at a distance H =^ Q feet below the constant 
level of the water J. and at a distance C = 4|- 



FiG. 780. 




feet below the top C of the pipe, 
substitute in the formula 

2 G / _ 



We must 



t =z 



V^gj* 



t = 



2 



fi^2 g . F 
^1 = 6, ^3 = 6 — 4,5 = 1,5, 

~pr — \^) — 144 and /j. = 0,81 ; thus we obtain 

144 / /- ,^^. 288 . 1,2248 



We - Vi,5) 



= 54,3 seconds. 



Fig. 781. 



0,81 . 8,025 ' ■ '"' 0,81 . 8,025 
§ 449. If the first vessel A B, Fig. 781, from which the water 

passes into the second, receives no water, and if its cross-section G^ 

cannot be considered as infinitely great, 
compared to the cross-section G of the 
other vessel C D, we must modify our cal- 
culation. If the variable distance Gi Oi of 
the first surface of the water above the 
level H R, at which the water in both ves- 
sels stands after the efflux, = x and the 
distance 6^ of the other surface of the 
water below the same, plane = y, the 

yariable head will \)Q x + y and the corresponding velocity of 

efflux will he V = V2 g {x + y), or, since the quantity of water 

GiX = Gy, 




v = Y2g{l +-|-J2/- 



The velocity with which the surface of the water rises in the sec- 
ond vessel is u J^ }^ F ./ ^ 1^ , G\ 

hence the corresponding retardation is 
'^Fx^i^ ^G 
G. 
58 



" = (i?)' (' - 1) '. 



914 



GENERAL PRINCIPLES OF MECHANICS. 



[§450. 



and the duration of efflux is 
t 



fjiF 



1 + 



"Z GVy 



|)^:(^)(l + |) 



Substituting instead of x and y the initial difference of level A, 

(C \ 
1 + -— ) ^ = A, we obtain 

h 

^ + -a- 

and the time in which the two surfaces of water come to one 
level is 



2 GVh 



2 G G, Vh 



Ixf{] 



G\ ,_ 



}jiF{G + G,) V29 



The time during which the difference of level changes from h 
to hi is, on the contrary, 

_ 2 G G,(Vn - VT,) 

~ fiF{G + G,) VY~g 
Example. — If the cross-section G^ of the vessel, from which the water 
flows into the other, is 10 square feet and the cross-section G of the vessel 
receiving the water is 4 square feet, if the initial difference of level be- 
tween the two surfaces of water is 3 feet, and if the cylindrical pipe which 
forms the communication is 1 inch in diameter, the time in which the two 
surfaces of water will reach the same level is 

3 . 10 . 4 . Vs 320 . 72 . V3 



0,82.8,025.^-.^ 



0,82 . 8,025 . 7 tt 



276 seconds. 



§ 450. Notch in the Side. — If the water issues through a 
notch, overfall or weir D F from a prismatic vessel A B C, Fig. 
Fm. 782. 782, into which there is no water dis- 

charged, the duration of the efflux is 
found in the following manner. Let us 
denote the cross-section of the vessel 
by G, the width F F of the notch by J, 
and the height B Fhjh, and let us di- 
vide the whole orifice of efflux into 
small strips, the length of each being 




§ 450.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 915 

r 

t and the height -. If the head is constant, the discharge per sec- 
III ' 

ond is 

C h 
dividing the contents of a layer of water by the latter, we oh- 

71/ 

tain the corresponding duration of the efflux 

Gil 



T = 



IfinhVig hf 



d G h * 

for which we will write . Ji—h 

2 [Ji n b V 2 g 
In order to obtain the duration ^ of the efflux of a quantity 
G [h — hi) or to determine the time during which the level of the 
water above the sill sinks from D U = li to D E^ = z^,, let us put 

^1 = — h, I.E. let A] consist of m parts, and let us substitute in the 

n 

last equation, instead of Ji -?, successively 

(fr.(=^''r'f^')"-(=:rr. 

and then add the results found. In this manner we obtain the re* 
quired time 



^ Gil 
t = 



2 II nh V 2 g 

S G h ll~^ r 3 / -. \ 3 3-\ 

-^, [m-l + (m + l)-t + . . . + n-^] 



2iinhV 2g ^^ 

= — --^^I^ [ (1-1 + 2-t + 3-1 + . . . + n-'A 
2iin-^l)V2q. 

'- (1-^ + 2-t -r 3-t + , . . + w-t) ], 

or, according to the Ingenieur, page 88, 

_ ^ G li~i / n-i_^^ _ m-t + 1 \ 

~ 2fin-ib V2'g \- I + 1 - i + 1/ 

fibV2g^^^ ^ J fibV2g^V hi V li 1 

If we ])ut 111 = 0, we obtain — z= and also i = 00; hence th« 

Vhi 

time required for the water to sink to the level of the sill will be 
infinitely great. 



91G GENERAL PRINCIPLES OF MECHANICS. [§451. 

Example. — If the water issues from a reservoir 110 feet long and 40 

feet wide, through an overfall 8 inches wide, in what time will the level 

Df the water fall from 15 inches above the sill to 6 inches above it ? Here 

we have 

3 . 110 . 40 / 1 1 \ 19800 



t =- 



. 40 / 1 1_\ _ 19800 _ 



'^'''' (1,4142 - 0,8944) = ^^^^^^^=}^^^ seconds. 



8,025 // ^ ' ' ^ 8,025 fi ^ 

If we assume as the coefficient of efflux fx = 0,60, we have for the real time 

of discharge 

• 1282 ^ 

t = — TT^^ = 2137,5 seconds = 35 minutes 37,5 seconds. 

Remakk. — For a rectangular orifice in the side we can put approxima- 
tively 

t = -^^ ((vi; - VT,) - 11 (vt;^ - v v^)), 

in which F and G denote the cross-sections of the orifice and of the vessel, 
a the height of the orifice, h^ the initial head, and h^ the head when the 

discharge ceases. If ^2 = 0' ^^^ orifice becomes a notch and the formula 

for overfalls must be employed. 

§ 451. Wedge-Shaped and Pyramidal Vessels. — If the 

vessel A B F, Fig. 783, from which the water is discharged, forms 

a horizontal triangular "prism, the time in 
Fig. 783. which it will empty itself is found in the 

/gizri~^^_ . yj B following manner. If we divide the height 
/^^^^^^^Jmn C E —Ji into n equal parts and pass hori- 
^^^^^^^^gSB * zontal planes through the points of divi- 
^^^^^^^H^ sion, the whole mass of water will be 
^^^^^^^f divided into equally thick horizontal layers,, 

||! whose common length i^A D = 1 and whose 

width diminishes from the surface down- 
wards. If the width D J5 of the upper layer = d, the width D^ B^ 
of another layer situated at the distance GEx — x above the orifice 

'X " 

F^ which is located at the lower edge of the prism, is y = j 5,, 

and its volume is yl . - = . But the discharge in the unit 

of time is 

Q = fi FV 2gx', 

h 
hence the small time, during which the water sinks a distance -, 

n 

hi X -n A/n h I 

T = ■ : fM FV2 g X = = . x^. 

^ fifiFV^g 



§ 451.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 917 



7/ 'H Tl 

FinallVj since the sum of all the x^ from :r = - to a; = — is 

n n 

we have the time of discharge of the whole prism of water 
il . , , . hi , , . 11 111 



n\i FV^g 



-=^,inU^l 



fiFV2g 
V 



.h^ = i 



fiFV^g h 



I.E. 



/ — 4 



in which F = h h IJi denotes the total volume of the water and 



Fig. 784. 




c = V%gh the initial velocity. In this case it requires ^ more 

time to empty the vessel than if the velocity c were constant. 

If the vessel A B F, Fig. 784, forms an upright paraholoidy we 

have the ratio of the radii K M — y and 
CD = h 

y __ Vx ^ 

^ ~ ~V%' 

hence the ratio of the horizontal section 

Gi through K to the base A D B = G i& 

G iP' oc 

■ —^ = ^ = -, and therefore 
G d^ h 

the volume of the layer of water is 

^ h Gx 
n n 
As this expression coincides exactly with that found for the trian- 
gular prism, we can put here also 

/ — 4 J ^ ^^ 
or, since F = ^ 6^ ^ (§ 124, Example), 

4_Z_ 



t 



This formula can also be employed in many other cases for the 
approximate determination of the duration of efflux, e.g., for de- 
termining the time required to empty a dam-. It is applicable, 
whenever the horizontal sections increase in the same ratio as the 
distances from the bottom. 



918 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 45L 



Fig. 785. 



If, finally, we have a pyramidal vessel 




B A B F, Fig. 785, to deal with, then 
Q^\ Q- = x" '. W, and, therefore, G^ = 

the volume of the layer Hi R^ is 

G, li _ G^ 



Gx' 



n 



and the time necessary to discharge the latter ia 



T r= 



Gx" 
n h 



: fj, FV2gx = 



G 



nil F h ^2 g 



• Jb'^m 



But since the sum of all the x^ from a; = - to i?; = — is 

n n 

ill 



/ l\i ni -J. 



we have the time necessary to empty the entire pyramid 



t^ 



G 



Gh^ 



— 6 



iGh 



or, putting I G h — V, 
V 



''i''^^-'*^FVYg~''l^^V2ir: 



— f) 



^ ~ '' ^ Fc 

Since in this case the initial velocity gradually diminishes from 
c to 0, the duration of the efflux is i greater than if the velocity 
remained invariable and equal to c. 

Example. — What time will a dam, the area of whose surface is 765000 
square feet, require to empty itself, when the discharge pipe enters at the 
deejjest place and is 15 inches in diameter and 50 feet long, and when the 
depth is 15 feet ? Theoretically 

V . . 765000 . 15 19584000 



t — 4 



— 4 1 



^gy. 8,025 VT5- ^■''''■^^^ 



gh 
= 200568 seconds. 
But the coefficient of resistance for the entrance of the water into the 
pipe, which is cut off at an angle of 45°, is 

C = 0,505 + 0,327 (see § 423) = 0,832, 
and the resistance of friction for the pipe is 



= 0,025 



I «2 50 .^2 ^2 



d ' 2g 



0,025 . 



• 2g 2g' 



hence the complete coefficient of efflux for the same is 

fi = -=== = — = = 0,594, 
Vl + 0,832 + 1 V2,833 

and the required duration of efflux is 

t = 200568 : 0,594 = 337655 seconds = 93 hours 47 minutes 35 seconds. 



§ 453.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 919 



Fig. 786. 



452. Spherical and Obelisk Shaped Vessels.— By the 

aid of the formulas, deduced in the foregoing 
paragraph, we can find the duration of the 
efflux from spherical, obelisk shaped, pyrami- 
dal, etc., vessels. 

1) The time required to empty a segment 
of a sphere A F B, Fig. 786, which is filled 
with water, whose radius C A = C F = r and 
whose height F G = h, is 




I, — 3 



n r ¥ 



nJi' 



1 o 



(10 r - 3 h) M 



or, if an entire sphere is to be emptied, in which case h = ^ r^ 

16 TT r' V¥r 



t — 



15 11 F ^Yg 



14 77 r^ Vr 



and for a hemisphere, where h — r, t = ^^ „ , . 

15^^4/2^ 

Here the horizontal layer H^^ R^ = G^, corresponding to the 

depth F Gx — X, is 

,^ . h 2 TV r h X n h x"^ 

— n X (2 r — X) 



n 



n 



n 



hence, if the velocity of efflux is v ^=^ ^2 g x, the duration of the 
discharge will be 

'0, TT f Jl. 

. x^ — 



rr 7i 



r z= 



Xi. 



n fj^F ^2g'^ ^ /^ ^ V¥g ' 
Since the first part of this expression coincides with the formula 
for the emptying of prismatic vessels and the second part with 
that for the emptying of pyramidal vessels, if we in the first 
case substitute 2 tt r 7^ for i I and in the second n h"" instead of G, 
we obtain by the aid of the difference of times required to empty a 
prismatic and a pyramidal vessel 

/ _ 2 III . and / - 2 _H 

^ ^ V2gh l^ ^ V2gn 

the time, in which a segment of a sphere 
will empty itself, as was found above. 

2) For a vessel A C K, Fig. 787, shaped 
like an obelislc or a pontoon, we can employ 
the above formulas ; for we can consider it 
to be composed of a parallelopipedon A E K, 
of two prisms BEN and DEN and of a 
pyramid (7 ^ iY (compare § 121). Let h be 



Fig. 787. 




920 GENERAL PRINCIPLES OF MECHANICS. [§45^. 

the width A D of the top, bi that K L of the bottom, I the length 
A B oi top, h that K N of the bottom and h the height F of 
the yesseL Then we have the surface A C of the water 

bl = bJr -r-b,{l- I,) + ?i (b - b,) + {I- I,) {b - h), 
in which bi li is the base of the parallelopipedon A E K, by {I — l^) 
and li {b — bi) the bases of the prisms BEN and D E K and 

(I - I,) (b - ^0 that of the pyramid C E JV, 
Fig. 788. Now the time required to empty the paral- 

^^- , ^ lelopipedon is 

\ \\ / // that required for the triangular prisms is 

\| ^ , _ ^ [br {I - ?0 +lr{b- h)] VI 

and finally that required to empty the pyra- 
mid is 

" ' l^F V¥g 

hence the time required to empty the entire vessel is 

V ^^^ fi "T fg I ^3 

= [30 5i/,+io b,{i-i,)+i(}i, {b-by)+e(i-k) (b-b,)] —-^^—= 

2 Vh 
= \fibl + Sb,h-^2 (b h + by I)] Tr^-~~=- 

15 fj, M ^2 g 

When Y — f ^^ vessel is a truncated pyramid. Putting in 
f\m case the base bl — G and the base bi h = G-\, we obtain 

t={^ G -¥^Gy + A:VGG,) ^-^^— .. 

16 fi F V%g 
It is easy to see that this formula will hold good for any trian- 
gular or polygonal pyramid. 

Example. — An obelisk-sliaped reservoir is 5 feet long and 3 feet wide on 
top and at a deptb of 4 feet, where a short pipe 1 inch in diameter and 3 
inches long is inserted in it, it is 4 feet long and 3 feet wide ; how long a 
time will be required for the water to sink 2|^ feet ? The time required to 
empty^ it is, assuming fx = 0,815, _ 

«= [8. 4. 2 + 3. 5. 3 + 2 (3. 4 + 5. 2)] ^ ^ , 

15 . 0,815 . 4 . (i^) • 8'^25 



§453.J EFFLUX OF WATER UNDER VAKIABLE PRESSURE. 921 



153 . 4 . 4 . 144 



= 153. 



2304 



= 153.7,475= 1144 sec 



15 . 0,815 . 8,025. tt 13,235 . 8,035 tt 
At the level 4 — 3f = 1|^ feet above the tube I = l^ + f = 4f and 
& = 5j + I = 3f feet ; hence the time required to empty the vessel, when 
it is filled to that height, is 

1 1 ^9, a/1 '^ 

e,=[8.4.2 + 3.^.^ + 2(2.^ + 4.^)].--^^--^^ 

= 603 seconds. 
The difference of these times gives the time (541 seconds) in winch the 
surface of the water will sink 2^ feet from the top. 

§ 453. Irregularly-shaped Vessels.— If we are required to 
find the duration of efflux for an irregularly-shaped vessel H F R, 

Fig. 789. 




Fig. 789, we must employ some method of approximation, such as 
Simpson's Eule. If we divide the whole quantity of water into 4 
equally thick layers and denote the heads corresponding to the 
different horizontal sections G^, Gi, G^, G^, G^, by Ji^, h^, li^, hz, h^, 
we obtain, according to Simpson's Rule, the duration of the efflux 



t = 



Ji^ — Tia 



(A 

\ 4/ 7, 



+ 



4.G, %G 



+ 



+ 



4 a. 



+ 



Ga 



V h. 



t = 



12nFV2g^Vh, Vh, Vji, Vh, 
If we divide it into six layers, we have 

K - h I G, ^ ^G, ^%G,^ ^G,^2G. ^G, 



)■ 



lSiiFV2g^Vh^ VT, Vh, 
The discharge in the first case is 



Gr 



VT, VJ, Vh, Vk 



;> 



V = 



F- 



K — h 

12 

K — h 

18 



(6^0 + 4: 6^1 -i- 2 G2 h 4z Gz -{- G^), and in the second 
(G, + 4: G, + 2G, + 4:G, + 2 G, + 4:G, + G,), 



If the form and size of the reservoir is not known, we can cal- 
culate the discharge Fby observing the heights Ji^, h-^, etc., of the 
water at equal intervals of time. If t is the whole duration of the 
efflux, we have for orifices in the side and bottom 

V = ^^Jl^^J ( i/T„ 4- 4 V^ + 2 4/T, + 4 ^¥3 + /T,), 
and for overfalls or notches 



V = 



12 



- ^2^( Vho' + 4 Fa? + 2 t^^ + 4 ^h' + Vh^'), 



6^3 = 410000 


(( 


G^ = 325000 


(( 


G^ = 265000 


u 



922 GENERAL PRINCIPLES OF MECHANICS. [§454. 

Example. — rin what time will the surface of the water of a dam sink 6 
feet, when the discharge-pipe is a semi-cylinder 18 inches wide, 9 inches 
high, and 60 feet long, and when the cross-sections of the surfaces of the 

water are 

for a head of 20 feet, G^ = 600000 square feet, 

" " 18,5 " G^ = 495000 " 

« " 17,0 " 

•« ■ " 15,5 " 

« " 14,0 " G^ = 265000 " ? 

Kow F =~ . (§y = ^ = 0,8836 square feet. If we put, as in the Ex- 

ample of § 451, the coefficient of resistance for the entrance of the water ia 

the pipe = 0,832 and that of the fricti(m, ^ 0,025 ^ ^ 0,025 . 60 . 1,091 = 

1,6356, we obtain the coefficient of efflux 

1 1 

u = - = — = 0,537, and 

Vl + 0,832 + 1,6356 V3,4685 

/^ F^2g = 0,537 . 0,8836 . 8,025 = 3,808. 
Now we have 

£. = m^l = 134170, ^ = *l^-5»» = 115090, 

^/h V20 VAj V18,5 

^ = !l^_» = 99440, A_ 835000 ^ ^^^^^^ 

VAg Vl7 V^g V15,5 

— ^ = =.- = 70830 ; hence the duration of the efflux is 

\/h^ Vl4 

t = ?-— - (134170 + 4 . 115090 + 2 . 99440 + 4 . 82550 + 70830) 

12 . 3,808 

1194440 

= 156833 seconds — 43 hours 33 minutes 53 seconds. 

"~ 7,616 

The discharge is 

Yz= ^ . (600000 + 4 . 495000 + 3 . 410000 + 4 . 325000 + 265000) 

= i^_??2^? = 2882500 cubic feet. 

2 

§ 454. Influx and Efflux. — If, while water is flowing out of 
the vessel, other water is flowing into it, the determination of the 
time in which the level of the water will rise or sink a certain dis- 
tance becomes much more complicated, and we are generally obliged 
to content ourselves with an approximate result. If the discharge 

per second into the vessel §i > fi F V2 gh, the water will rise, and 

if §1 <C. fi FV2g h, it will sink. But the level of the water be- 
comes constant, when the head is increased or diminished, until it 

becomes equal to ^ = ^ ( -^ ) . The time t, during which the 

t7 ( 



§ 454.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 923 

variable head x is increased a small quantity |, is determined by the 
equation Gi^ — QxT — \i F V%~gx . r, 

and, on the contrary, the time, in which the surface of the water 
sinks a distance ^, is determined by the equation 

G, ^ = II FV¥g~x .T ~ Q,T. 
Hence we have in the first case 

T = ^ — , and in the second 

Q,- fiFV2gx 

^^ _G'i^ 

fiFV2jx- qI 
By employing Simpson's Eule, we obtain the time of discharge, 
during which G^ becomes successively G^, G.^, etc., and the head h^ 
becomes hi, Ji^, . . ., 
^ ^ K-K r ^0 ^ 4 G, ^ 2 G, 



4 (?3 G^ 

+ := + 



■} 



liF\/%g}i,- ft ^FV^gh,- ft- 

or if we denote ^— =: by V k ,wq have more simply 

fi F V^ g 
f = ^0 - fh r ^0 _^ 4 6^1 ^ 2 G, 

12 fi F V2g L i/T, - VT 4/1; - VT "^ 4/T2 - VI^ 



4 G, ^ G,_ 

VTs- VT VT,- Vic 



+ -=^-^-= + 



] 



If the vessel is prismatic and its cross-section is constant 
and = G, we have (see the author's " Experimentalhydraulik," 
§ 9, XII) 



2 G r ,_ ^ • (Vli-Vk^ 



for the time, in which the head h changes to hi. 

^. ^ , , Vh- Vk Vh - Vk 
. b]nce for hi = k, — — — = = 00 , 

Vh- Vk 

it follows that the level of the water becomes permanent after an 
infinite time has elapsed. 

For a notch in the side we have the following formula 

_ G_^ [1 ( Vh - Vky (hi + VhTk + k) 

^ ^k+ {2Vh+ Vk){^Vhi'^ Vky 



924 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 455. 



in which Ic = ( ^ — y^—= I and I denotes the Naperian losfarithm 

and tangr^ y the arc whose tangent \& y. 

According as y^ § 7i or the discharge into the vessel 

a rising or a sinking of the water in the vessel takes place. The 
state of permanency occurs, when lix = k, but in this case the cor- 
responding time t becomes = oo . 

Example. — In what time will the water in a parallelopipedical box 
13 feet long and 6 feet wide rise 2 feet above the sill of a notch in the side 
^ foot wide, when the discharge into it is 5 cubic feet per second ? Here 
we have ^ = and consequently more simply 

Now G =:12.Q = 72 feet, Q^ = 5, h^ = 2,i = ^, fi = 0,6, and 
= (lTo;6TT78;035J = ^'^^^' 



:]■ 



Jc 



hence the time required is 



t = 



72 . 2,1330 



3. 5 



I 



4,1330 + V4,2660 



Ve 



= 10,238 



I 



(1,4142 
6,1984 



- 1,4605) 

Vl2 tang, 



)] 



— 1 



L 0,002144 
= 10,238 (7,969 - 1,776) = 10,238 



- ^^^ ^^^ " (i;4142 + 2,9210 

W6 X-l 

\4,8352/J 
, G,193 = 63^ seconds. 

-We can make a useful applica- 



§ 455. Locks and Sluices. 

tion of the principles just enunciated to the filling and em^Jtying 
of lochs and sluices (Fr. ecluses ; Ger. Schleusen). We distinguish 

two kinds of locks, name- 
ly, the single and the 
double. The single loch 
consists of a chamber B, 
Fig. 790, which is sepa- 
rated from the water in 
the head lay A by the 
gate H F and from that 
in the tail hay C by the 
gate R 8. The double 
loch, Fig. 791, on the 
contrary, consists of two 
chambers with an upper 
gate K L,?i middle one 
HF, and a lower one R 8. 




Fig. 791. 




^ 455.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 925 

1) If we put the mean horizontal cross-section of the chamber 
of a single lock = G, the distance 0^ of the centre of the open- 
ing in the upper gate below the surface H R of the water in the 
head bay = h^, its distance Ox Oo above the water in the tail bay = h^ 
and the cross-section of the orifice in the gate = F, we have the 
time necessary to fill the lock to the middle of the orifice, during 
which the head is constant, 



tx = 



Gh2 



and the time necessary to fill the remaining space, during which 
there is a gradual diminution of head, 

_ 2 Ghx 

' ~ ^ ^ V2gTx' 
hence the time required to fill the whole lock is 
/ _ , ^ / _ (2 Jh + h) G 
l^^ V2gh, 
Tf the orifice in the lower gate is entirely submerged, the head 
decreases gradually during the emptying from O2 = Ai + ^a to 
zero, and the time of emptying the lock is, therefore, 

. _2G Vh, + h, 
— 



H'J^i''2g 

But if, on the contrary, a part of the orifice lies above the 
lower water level, we have to consider two quantities of water, one 
discharged above and the other below the water. Putting the 
height of the portion of the orifice above the water = a^, the height 
of that below the water = 0^2 and the width of the orifice = h, we 
obtain the duration of the efl&ux by means of the formula 
, 2G{7h±Jh) 

F ^ V2g («i \/h, + A, - -|i + «!2 V'^r+^2) 

2) In the double lock (Fig. 791) the head in the upper cham- 
ber which is cut off from the head bay gradually diminishes during 
the efflux into the second chamber. If G is the horizontal cross- 
section of the first chamber, and if the initial head Oi = h^ 
is diminished to X 0, = x, while the water in the lower chamber 
rises to the middle of the orifice in the gate a distance Oi 0^ = h^, 
wo have the time corresponding to it 

'• = ;,r^v^ ( ^ - '^)- 

But the discharge is 



926 GENERAL PRINCIPLES OF MECHANICS. [§ 45^. 

G {hi — x) = G^hii hence 

X = hi — -^ hi and 
G 



ti = 



2 G 



2 V'^ 



(vj;-/^^) 



{VGhi - VGh, - GJio). 



The time in which the water in the second chamber rises to a 
/evel with that in the first, or in which the water in the two cham- 
bers assumes a common level, is, according to § 449, 

' _ 2GGiV~x _%Gi\/~G VGhi - Gih, 



liF{G+Gi)V¥~g \^F{G^Gi)s/Yg ' 

and the whole time required to fill it is 

t^ti^U= \^^— ( ^^^ - -rA-n ^(^h- Gi h\ 

Example. — What time is necessary to empty and fill a single lock of 
the following dimensions : mean length of the lock = 200 feet, mean width 
= 24 feet or G^ = 200 . 24 = 4800 square feet ; distance of the centre of 
the orifice in the upper gate from both surfaces of water = 5 feet, width 
of both orifices = 2^ feet, height of the orifice in the upper gate = 4 
feet, and height of the orifice (entirely submerged) of the lower gate = 5 
feet ? Substituting in the formula 

^ ^ (2 h, + h ,) G 

fxF^2 gh 

y^^ = 5, /^g = 5, 6^ = 4800, II = 0,615, i^ = 4 . 2| = 10 and ^/2g = 8,025, 

we have for the time required to fill it 

3.5.4800 14400 ^^^^ _ . ^^^ 

f _ — n = 6524 sec. = 10 mm. 52|- sec. 

~ 6,15 . 8,025 V5 1,23 . 8,025 V5 

If we substitute in the formula 

t = 2G^h^+Ji^ ^ ^ ^ ^g^^^ A^ 4- Ag = 10, i^ = 5 . 21 = 12,5, we have 

fx F'\/2 g 
the time necessary to empty the lock 

2 . 4800 VTO ,^^ -, ^ . . w« 

t = ^ ^^^ — TTT^ — TT^^Tzi:: = 492 sccouds = 8 mmutcs 12 seconds. 
0,615 . 12,5 . 8,025 

§ 456. Apparatus for Hydraulic Experiments.— By means 
of the apparatus represented in Fig. 792, we can not only show by 
more than 100 experiments the most important phenomena of 
efflux, but also prove in figures the most important of its laws. 
The apparatus consists of a discharging vessel ABC, which is 
provided with three orifices Fi, F<^, Fl, whose centres are at dis- 



§456.] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 927 

tances from the mean level of the water, which are to each other 
as the squares 1, 4, 9. To these orifices various mouth-pieces and 
pipes can be applied, and in order to do this without being dis- 

FiG. 792. 

A A 




turbed by the water, we close the orifice by means of a particular 
kind of valve H^, Hz, to which is attached a rod passing through a 
stufl&ng box in the back of the apparatus. In the upper and wider 
part A B oi the apparatus two pointers Z^ and Z^, which are 
directed upwards, are placed. These serve as fixed points, the one 
marking the beginning and the other the end of the experiment. 
The water which is discharged is caught in a vessel, which before 
each experiment is placed on top the discharging reservoir, into 
which its contents are emptied by opening an orifice that is gen- 
erally closed by a stopper. 

In order to find by the aid of this apparatus the coefficient of 
efflux y, for diiferent mouth-pieces and tubes, we must observe by 
means of a good stop-watch the time t, in which the water-level 
sinks from one pointer to the other, or within which the head 7i, 
becomes h^] if, then, i^is the cross-section of the orifice and G the 



928 GENERAL PRINCIPLES OF MECHANICS. [§456. 

area of the sinking surface of the water, we have the coefficient of 
efflux (see § 448) 

2 G ( \/h. - Vh,) 
a z=z ^^ -J 

FtV%g 
and the corresponding mean head 



h = 



/ V lh + vw 



This apparatus is provided with a collection of mouth-pieces and 
tubes, viz., square, rectangular, circular and triangular orifices in a 
thin plate with or without an internal rim, short cylindrical and 
conical tubes, long straight tubes of different diameters, elbows, 
bends, etc., which can be inserted in the different openings F^, i^, 
jPg. By means of an apparatus with the above accessories we can 
show in a few hours all the phenomena and laws of efflux ; with it 
we can study not only the perfect and imperfect and complete and 
incomplete contraction, but also the different degrees of the con- 
traction of the jet, and we can make ourselves acquainted with the 
resistance of friction, with that of elbows and bends, and also, by 
observing jets of water and the sucking up of water, with the 
positive and negative pressure of water. We will always find 
results which agree pretty well, and sometimes extraordinarily 
well, with the coefficients given by experiment (ju, 0, a, (^). In our 
apparatus G = 0,125 square meters, the usual diameter of the 
orifices and tubes is 1 centimeter, and for the lower orifice 
^1 — 0,96 meters and h^ = 0,84 meters. (A detailed description of 
this apparatus and of the experiments, etc., which can be made 
with it, is given in the author's " Experimentalhydraulik.") 

The following example shows how well observations with this 
apparatus agree with the well-known experiments on a large scale. 
With a short cyhndrical tube placed in the lower aperture, t was 

= 33, and with a long glass tube, for which the ratio -^ = 124, t 

was found to be = 56 ; from this we deduce in the one case 

1 

Ml 



Ml = 0,815 and ^1 = -i^ - 1 = 0,504, 



and in the other 

1^2 - 0,480 and ^'s = — - 1 = 3,332; 

hence 

^2 - ^j = 3,332 -- 0,504 = 2,828, 



§ 456] EFFLUX OF WATER UNDER VARIABLE PRESSURE. 929 
and therefore the coefficient of friction for the tube is 

^ = ^ (f, _ <,) = ^ig, = o,om 

According to the first table in § 429, for the mean velocity v = 
1,84 meters, with which the water was discharged from the tube, 
4" = 0,0215 ; the results agree, therefore, very well. By means 
of these experiments, we can satisfy ourselves that the velocity of 
efflux of the water does not depend at all upon the inclination of 
the tube, but upon the head of water above the orifice of discharge. 
The duration of efflux is the same, no matter whether the long tube 
is inserted in the lower or middle opening, provided its orifice of 
discharge is at the same depth below the surface of the water in the 
reservoir. 

This apparatus has recently received many additions, so that we 
can now make with it experiments upon the efflux of water under 
constant pressure, upon the efflux of air, and also upon the pressure, 
impact, and reaction of water. 

Closing Remark. — A very complete list of the works upon the subject 
of e£B.ux of water and upon the motion of water in tubes is given in the 
" Allgemeine Machinenencyclopadie," Yol. I, Art. '' Ausfluss." We will 
mention here, among the later works, Gerstner's " Handbuch der Me- 
chanik," Vol. 2, Prague, 1833 ; d'Aubuisson's " Traite d'Hydraulique a 
Tusage des Ligenieurs," II edit. 1840 ; Eytelwein's " Handbuch der Me- 
chanik fester Korper und der Hydraulik," 3d edition, 1843; Scheffler's 
" Principien der Hydrostatik und Hydraulik," Braunschweig, 1847. The 
older works of Bossut and du Buat upon hydraulics are always of value on 
account of their practical treatment of the subject. "Die Experimental- 
hydraulik, eine Anleitung zur Ausfiihrung hydraulischer Versuche im 
kieinen," by J. Weisbach, Freiberg, 1855, is particularly adapted for teach- 
ing and for the practical study of hydraulics. Riihlmann's " Hydrome- 
chanik" is also to be recommended. The more recent works of Lesbros, 
Boileau, Francis, etc., have been mentioned before (§§ 378, 880 and 387). 
"We can also recommend Rankine's "Manual of Applied Mechanics," as 
well as Bresse's " Cours de Mecanique Appliquee," 11. But two parts of 
the hydraulic experiments of the author have as yet appeared, and they are 

1) " Experiments upon the efflux of water through valve-gates, cocks, 
clacks, and valves ;" and 

2) "Experiments upon the incomplete contraction of water during 
efflux, etc., Leipzig, 1843." 

Several new treatises by the author upon hydraulics are contained in 
the " Civilingenieur," the " Zeitschrift des Deutschen Ingenieurvereines," 
■ etc. . .» . iiw -. ', 

59 



930 GENERAL PRINCIPLES OF MECHANICS. [§457. 



CHAPTER VI. 

OF THE EFFLUX OF THE AIR AND OTHER FLUIDS FROM VESSELS 

AND PIPES. 

§ 457. EfQux of Mercury and Oil. — The general formula 

V = V¥gli (see § 397) 

for the velocity v of efflux of water under a pressure, measured by 
the head h, holds good (see § 399) also for other liquids, such as 
quicksilver, oil, alcohol, etc., and can also be employed for the ef- 
flux of air and other aeriform fluids, when the pressure is not very 
great. If y denotes the heaviness of the fluid and p its pressure 

upon the unit of surface, we have in like manner A = -^ and 

therefore 



V = j/2 1 



9 -' 

7 

If we measure the pressure by means of a piezometer, filled with 
a liquid whose density is yi, the height of the column of liquid is 

hence p = Jii yi, and therefore 



in which e, = — denotes the ratio of the heaviness of the liquid in 

r 

the piezometer to that of the fluid which is being discharged. 

This agreement of the laws of efflux for different fluids is not 
confined to the velocity alone, but extends to the contraction of 
the fluid vein. Streams of mercury, oil, air, etc., when passing 
through an orifice in a thin plate, are contracted in almost exactly 
the same manner as a stream of water. Some experiments made 
by the author upon the efflux of mercury, oil and air, have shown 
conclusively this agreement (see the Poly tech n. Centralblatt, year 
1851, page 386). These experiments gave 

1) With a circular OTi^cQ in a thin plate 6,5 millimeters in di- 



§457.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 931 

ameter, under heads of 91,5 millimeters and 329 millimeters, the 
coefficients of efflux 



For water. 


Mercury. 


Rape-seed oil. 


[JL = 0,709 


0,670 


0,674 



From the above table it appears that the contraction of streams 
of mercury and rape-seed oil is a little greater than that of a stream 
of water. 

2) With a short, well-rounded, conoidal mouth-piece, whose di- 
ameter d was Q,Q m. m. and whose length was double the diameter 
{I — 2 d), the following values were found 



For water. 


Mercury. 


Rape-seed oil. 


At a temp. i2i° C. 


1 

At a temp. 39° C. 


11 --= 0,942 


0,989 


0,430 


0,665 



3) A short cylindrical pipe, which was not rounded off inside, 
whose diameter was d = 6,76 millimeters and which was three 
times as long as wide (1 = 3 d), gave the following values : 



For water. 


Mercurj'. 


. Rape-seed oil. 


At a temp. 12^° C. 


At a temp. 39° C. 


VL = 0,885 


0,900 


0,363 


0,604 



From these experiments we find that mercury flows through 
short mouth-pieces and pipes but little faster than water, and that, 
on the contrary, the velocity of rape-seed oil increases visibly with 
the temperature and is less than that of water. The great differ- 
ence between the velocity of water and oil is due to the greater ad- 
hesion of the oil to the walls of the pipe. 

4) The following values of the coefficient of resistance C were 
obtained with a glass tube 6,64 millimeters in diameter and 86 
times as long as wide (I) and with an iron tube 6,78 millimeters in 
diameter and 85 times as long as wide (II). 



932 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 458. 



1 

i 

1 

1 

I. 
II. 


For water. 


Mercury. 


Rape-seed oil. 


At a temp. i2i° C. 


At a temp. 39° C. 


^ := 0,0271 


0,0277 


39,21 


2,722 


4- = 0,0403 


0,0461 


54,90 


5,24 



According to this last experiment the coefficient of resistance 
of mercury in an iron or glass tube is a little greater, and, on the 
contrary, that of rape-seed oil many times greater than that of 
water. We also see from these tables that the coefficient of resist- 
ance of the rape-seed oil diminishes as the temperature or degree of 
fluidity increases. These experiments also show that the coefficient 
of resistance for the iron tube is much greater than for the glass 
tube, which is due to the greater smoothness of the latter. 

§458. Velocity of Efflux of Air. — If we assume that i^^ 
air does not change its density during the efflux, the well-known 
formula for the efflux of water from vessels can also be applied to 

the efflux of air. If p is the pressure of the 
exterior air and p^ and 71 the pressure and 
heaviness of the air inside the vessel A B, 
Fig. 793, we can put for the velocity of ef- 
flux of the latter (see § 399) 



Fig. 793. 
M 




■ ^ Ti 

-^ Ti \ pJ 



But (according to § 393), if J9 is the pressure in kilograms upon 
a square centimeter of surface, y the weight of a cubic meter of air, 
and T its temperature 

p _ 1 + 0,00367 . r 
r ~ 1,2514 ' 

or, ifp is referred to a surface of one square meter, 

p _ 10000 
y ~ 1,2514 
hence it follows that 



(1 + 0,00367 r) = 7991 (1 -1- 0,00367 r) ; 



§459.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 933 
Vz^ = V^ = 4/7991 Vl + 0,00367T, ' 



71 ' r 

or replacing 0,00367 by 6 



y^ = 89,39 Vl -\- 6t, and v = 89,39 \/2 g{l -\- 6 r) (l- ^\ 



Pi 

396 



i/(i + <5.)(i-A 



P^ 
or for the English system of measures 



V = 161,9 y 2 (/(I + S t) /l - ^\ 

= 1299 |/' (1 + (5 r) (1 - ^\ 

T being expressed in degrees of the centigrade thermometer. 

If ^ is the height of the barometer and h that of the manom- 
eter (M), we have also 

p h ^ p Ji 

^^ — -, or 1 — ^ — 



and consequently the velocity of the issuing air 



V - 396 V (1 + (5 r) J meters 



= 1299 i/ (1 + 6r) - A^ feet, 

or approximatively, when the height of the manometer is small, by 
putting 

1 =1- ^ 



l/lH- , 



I 3*' 



t) = 396 /l - ^) V (1 + <5 t) ^ meters 



= 1299 (1 - A) ^'(1 + 5 r) I feet. 

Remakk. — On account of the ordinary humidity of the atmosphere, it is 
advisable in practice to take f5 = 0,004. 

§ 459. Discharge. — If F is the cross-section of the orifice, we 
have the effective discharge, measured at the pressure in the reser- 
voir, pi or i + h, 



934 GENERAL PRINCIPLES OF MECHANICS. [§460. 






^/2,V. ' 



E.G., for atmospheric air 



0, ^ 396 7^ r ^^-^-^^ cubic meters 

= 1299 i^ '/ii^--lIZ' cubic feet. 

If we reduce this quantity of air to the pressure of the exterior 
air p or &, we obtain 

y p ih 



E.G., for atmospheric air 

§ = 396 i^ y (1 4- (^ r) /l + I) I cubic meters. 

e = 1299 F y{l + ^ ^) (l + ^) ^ cubic feet. 

Example. — The air in a large reservoir is at a temj)erature of 120° C 
and at a pressure corresponding to a height of the manometer of 5 inches, 
while the barometer marks 29,3 inches ; what will be the discharge through 
an orifice 1^ inches in diameter ? 

The theoretical velocity of efflux is 

V = 1299 y (1 + 0,00367 . 120) -^ = 1299 |/ — y^ y- = 596 feet, and 

the cross-section of the orifice is 

F=-j- =-^. (gj = 25g = 0,01227 square feet; 

hence the theoretical discharge, measured at the pressure in the reservoir, is 

Q^ = Fv = 596 . 0,01227 = 7,313 cubic feet, 
and, on the contrary, at the exterior pressure the volume is 

Q = ^-^ Q = 11'^ . 7,313 = 8,565 cubic feet. 

§ 460. Efflux according to Mariotte's Law. — If we suppose 
that the air does not change its temperature during the discharge, 



§ 460.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 935 

we can assume that it expands according to the law of Mariotte 
(see § 387), and therefore that the quantity of air Q in passing from 

the pressure^ to the pressure ^1 performs the work Q p 1 1~-\ If 

we put this work equal to the energy - — Q y stored by § 7 during 
the efflux, we obtain the following formula 

2g y \pj' 
hence the velocity of efflux is 

Now, as in the foregoing paragraph, for the metrical system of 
p _1 +61 
y ~ 1,2514 



V 



measures - = ^ . : hence we have here also 



V = 396 y (1 + 6r)l U^A = 396 \/ {1 + S r) I (^-) meters, 



^P 
and 



V = 1299 \/{l + 6r)l{^^\ = 1299 |/(1 + 6t)1 ^-^-) feet, , 

in which b denotes the height of the barometer in the exterior air 
and h the height of the manometer for the confined air, r the tem- 
perature of the latter in degrees centigrade and 6 = 0,00367 the 
well-known coefficient of expansion of air. Now the theoretical 
discharge per second is 



Q = Fv = F^2gp[^) 



= 1299 F \/{l + (5 r) ? (^-t--) cubic feet, 
or, when reduced to the pressure of the air in the reservoir, 

Pi Px y \pf 



h 



-^<^m 



b -}- h ^ -^ y 



= imF^y^l,Sr^l(^^^. 



936 GENERAL PRINCIPLES OF MECHANICS. [§461. 

h 

If the excess of pressure of the air in the reservoir, or t> is very 
small, we can put 

(see the Ingenieur, page 81), and therefore, approximatively, 



,= V^,|(i-i^)f, 



r 

while according to the first formula for the efflux (see § 459) 

We see that if we assume that air in flowing out expands 

according to Mariotte^s law, we obtain a smaller discharge than 

when we consider that the air acts exactly like water and does not 

. h 
expand at all. This difference diminishes with ^ and in both cases 

for very small values of y, we have 

Q = Fy2g^, I = 1299 Fy(l + <^ t) | cubic feet. 

§ 461. Work Done by the Heat. — The logarithmic expres- 
sion, found in § 388, for the work done during the compression or 
expansion of air is correct only, when we assume that, while the 
change of volume or density is taking place, the temperature of 
the air does not alter ; but this is correct only, when the change 
takes place so slowly that the heat in the confined air has time 
enough to communicate any excess to the walls of the vessel and 
to the exterior air. But if the change of density takes place so 
quickly that it is accompanied by a change of temperature, when 
the air is compressed, the temperature is elevated and when it is 
expanded, it is lowered. Under these circumstances the tension 
cannot change according to the law of Mariotte alone. If p and 
pi are the pressures, y and y^ the heavinesses and r and Tj the tem- 
peratures of the same air, we have, according to § 392, the formula 

Pr _ 1 -\r 6 T , y^ 

p 1 -h 6 T ' y' 

Now if during the sudden change of pressure the temperature 
varies in the ratio 

1 ^ St \y/' 



§461.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 937 

we can put 

p \l + 6r/ \y)' 



or 



7}^ /i±_iiiV_/M^ 

y Kl-hdrl U/' 



Fig. 794. 



y \1 -\- 6 T / \p 

If in a cylinder A G, Fig. 794, a prism of air, whose initial 
height is U B = s, whose initial tension is p 
and whose heaviness is y, is cut off by a piston 
B F, and if, by suddenly raising the piston a 
distance x, we cause the density of this mass 
of air to become y and its tension to become 
Zj we have, according to the last formula, 

^ = (y)^= ( ' f 
p \y/ \s — xl ' 

and therefore 

^ ~ \s + x) ^' 
In order to move the piston, whose area 
we will for simplicity put equal to the unit 
of surface, through an element a of its path 
the work, which must be done, is 




o = {^yJP o =pasl {s - x)- 



2. 



Substituting instead of x successively 1 o, 2 o, 3 a . . . and put- 
ting s = n o and the height of the prism of air, when the piston 
has described the space E E^, Ex B — s^ — m g, we have for the 
work done by the piston in moving the distance E E^ 

Ai= p o s" [s-" + (5 — (T)-f -f (5 — 2 (7)-t 4- . . . + (5 — m (7)-t] 

(■ 



__p s^ j 1-t + 2-t + 3-t + . . . + m-l + . . . + n- 
~ "(^ I - (1-1 + 2-t + 3-t + . . . + m-t) 



Now, according to page 88 of the Ingenieur, when m and n are infi- 
nitely great numbers, we have 

1 s ^ a o s 3 ^~* ^ 

1-1 + 2-t + 3-t + ... -f M-t = r = 7 

— A m* 

and '^ - 



938 GENERAL PRINCIPLES 07 MECHANICS. [§461. 

1-f + 2-t + 3-1 + . . . + ^-^ = ~; 

hence 

If by raising the piston another distance s we wish to force the 
compressed mass of air A E^ into a space R, where the pressure is 

the work to be done will be 

^2 — P\ Sx — —- ; 

the exterior air presses upon the piston during the whole of its 
course with a force p and transmits to it the mechanical effect 
Az = 2^ s. Hence the total mechanical effect necessary to compress 
the Yolume of air (1 . s) and force it into the space R is 

and consequently the work done in compressing a volume of air 
from the pressure p to jh is 

while, according to Mariotte's law, we should put 
and for perfectly incompressible fluids we have 

If, on the contrary, the quantity Fj y^ of air at the pressure jy, 
is brought back by sudden expansion to the pressure p and the 
density 

or to the volume 

the work done by air is 



§462.] EFFLUX OF THE AIR AND OTHER FLUmS, ETC. 939 

Example. — If a blowing engine converts per second 10 cubic feet of 
air at a pressure 5 = 38 inches of the barometer into a blast at the pressure 
h + h = SO inches, it requires, according to the formula, 

since the pressure per square foot is 

jp = 144 . 0,4913 5 = 144 . 0,4913 . 28 = 1981 pounds, 
the mechanical effect 

^ = 30 . 1981 (y 2g - l) = 59430 [\/\i -A = ^^^^ • ^^2326 

= 1382 foot-pounds. 
The logarithmic formula (see Example 1, § 388) gives A = 1366,7 foot- 
pounds, and that for water 

A = Vp (^^ - ^ ^^^^^ (n ~ ^) ^ ~S^ "" 1415 foot-pounds. 

§ 462. Efflux of Air, when the Cooling is taken into 
consideration. — The energy A = 3 Q^ ^j^ 1 — (—) I which is 
restored during the sudden expansion of Q^ to Q, can be put equal 

to the work Qi yi . ^— done in overcoming the inertia of the mass 

^^ 

y 

of air when the latter assumes the velocity v. 

From the equation 

we deduce the following formula for efi9.ux : 
2g yi L \pi/ J 



hence we have in meters 



^=^'^-n'-&}' 



V = 154,8 1/2 5- (1 + <Sr)[l - (|-)*] 



Pi 

= 685,8 4/(1 + S r) [l - g_)*] 
and in English feet 



V = 280,4 \/2g(l + (5 r) [l - (|-)*] 
= 2250 y (1 -f cJ r) [1 - /^V] feet 



940 GENERAL PRINCIPLES OF MECHANICS. [§462. 

The tension of the issuing air is that of the exterior air^; its 
heaviness is / p\i 

and its temperature is (PV -i 

and the theoretical discharge from an orifice, whose area is F, is 



^--^^ = ^i/^[^-©1 



= 280,4 i^y 2 g {1 + d t)\i ~ {—T\ cubic feet, 

in which ^i, ji and Tj denote the pressure, heaviness and tempera- 
ture of the confined air. " 

Eeduced to the pressure in the reservoir, this discharge is 

«• = 7; ^-g;)'*-^©^^^--^ [■-©'} 

and, finally, reduced to the pressure of the exterior air and to the 
temperature of the air in the vessel or to the heaviness y = 

y. (|;),itis 



K we put — = — 7 — , in which h denotes the height of the bar- 

rometer in the exterior air and h -{- h that of the barometer in the 
confined air, we obtain 

= 280,4 V^,(l+dx)(^^)*[(^)»- 1] 

= 3250 F\/{l + d r) (^y [/L±i\*_ ij cubic feet 
In most cases T is "^ery small, and we can put 

m*-(-i)'='-it-(tr-- 



§463.j EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 941 



and therefore 






=^[x-^.gji/..f:l 



In the application of this formula to fans, blowing engines, etc., 
in which cases j < -], the theoretical discharge, measured at the ex' 
terior pressure and the interior temperature, is simply 

= 89,39 Fy^g{l + dr)| =r 396 F y (1 + 6 r) | cubic meters 

= 161,9 Fy^g{l + 6r)^ = 1299 Fy {1 + ^ '')\ cubic feet. 

Example. — In the case treated in the Example of § 459, where 5 = 29,2, 

h = 5 inches, r = 120° and F = ^- = 0,01227 square feet, we have the 

discharge according to the last formula, measured at the pressure of the 
external air, 

Q = 1299 F i/ 1,4404 . -^ = 1299 F V0,2466 

= 645,1 F = 645,1 . 0,01227 = 7,915 cubic feet, 

while previously (§ 459) we found, according to the formula for water, 

Q = 8,565 cubic feet, and according to the logarithmic formula in § 460, 

we have 

/ 34 2 

Q = 1299 F y 1,4404 Z—'- = 1299 i^ V0,2277 

= 619,9 . 0,01227 = 7,606 cubic feet. 

§ 463. EflElux of Moving Air.— The formulas for efiSux 
already found are based upon the supposition that the pressure p 
or the height h of the manometer is measured at a place, where the 
air is at rest or moving very slowly ; but if we measure pi and hi 
at a point, where the air is in motion, if, e.g., the manometer M^ 
is in communication with the air in a pipe (7 i^, Fig. 795, we must 



942. 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 463. 



take into consideration, in determining the velocity of efflux, the 
vis viva of the approaching air. If c be the velocity of the air pass- 
ing the orifice of the manometer, we must put 



"A 1 

Gi Ti S-: = Vi Ti 



. 3 ft.. [1 .-(!;/] 



2(/ ^^ " 2^ L \p. 

If F denotes the cross-section of the orifice and G that of the 
tube or of the stream, which passes the orifice of the manometer, 
the discharge of air is (Ji 7i = G c jx — F vy^^', hence 

c_ _ Fy^ _ F^ 
'G 



V 



Oy, 



©'"* 



e- ['-©• (t)'] f; = "«-['- ft)']. 



Ihi J 2 ^ 
and the required velocity of efflux is 



V = 







Pi 

or approximatively, when p^ is not much greater than p^ 



V = 




'-(f)' 



= 2250 




' - (IF 



feet. 



Here, as in the case of the efflux of water, the velocity of efflux 

F 



Fig. 795. 



increases with the ratio ^ of 

the cross-section of the ori- 
fice to that G of the pipe or 
moving stream of air. We see * 
from this that, under the same 
circumstances, the height p^ 
of the manometer decreases 
as the diameter of the tube 
diminishes, or as the velocity of the air in the pipe increases. 




§463.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 943 

If we denote by ih ^^^ tension in the reservoir, where the air is 
at rest, we have also 

and if we eliminate v from the two expressions, we obtain 



y/ = 1 



If l denotes the height of the barometer in the free air, h that 
of the manometer connected with the reservoir and i^the area of 
the orifice of efflux, we have, finally, the theoretical discharge, 
measured when its heaviness is 




Q = F\ 'j^rr. = 161^9 F 




1299 F 




2^(l + cJr)| 

HIT 

(1 + <S r) I 



(i1 



Example. — The height of a quicksilver manometer, which is placed 
upon a pipe 3| inches in diameter through which air is passing, is 2^ inches, 
while the air is discharged through a circular orifice 2 inches in diameter 
at the end of the pipe : what is the velocity of discharge, assuming the 
barometer in the external air to stand at 27|- inches and the air in the pipe 
to be at a temperature of 10° C ? Here 

Vl + (5r = VT,0367 = 1,018, j/v = V^ = V^ = 0,3015 and 
F= Tvr'' = 3,141 : 144 = 0,02181 and 



/-(f-y 



» V4r - 16' 46,814 „„,„ 
;--^^t^—---V = 0,9452; 

hence the discharge is 



q = 1299 F. i5Mj_5J215 ^ 421,8 i^= 9,20 cubic feet. 
For the corresponding tension p^ in the reservoir, we have 

0,0287 „„„„,„ ,_ 
= 078934 = "'"^'1^ • ^^°<=« 



944 GENERAL PRINCIPLES OF MECHANICS. [§464 



r n 



= 0,90788, j9o = 1,103 j9 and h + \ = 1,103 h 

and consequently the height of the manometer in the reservoir is 
\ = 0,103 5 = 0,103 . 27,5 = 3,83 inches. 

§ 464. Coefficients of Efflux. — The phenomena of contrac- 
tion^ which we have studied for the eflfiux of water, are also met 
with in the efflux of air from vessels. If the orifice of efflux is in 
a thin plate, the stream of air has a smaller cross-section than the 
orifice, and the effective discharge ft is consequently smaller than 
the theoretical Q, or the product F v oi the cross-section F of the 
orifice and the theoretical velocity v. This diminution of the dis- 
charge is owing principally, as we can observe in a stream of 
smoke, to the contraction of the stream of air, and we can, there- 

F 
fore, as in the case of water (see § 406), call the ratio a = — of 

the cross-section i^i of the stream of air to that F of the orifice 

the coefficient of contraction, 

the ratio </> = — of the effective Telocity Vx to the theoretical v 

(see § 408) 

the coefficient of velocity. 

and the ratio // = ^ = -~-^ = a (p of the effective discharge ft 

to the theoretical discharge Q 

the coefficient of efflux. 
As in the case of water the coefficient of velocity for the ef- 
flux of air through an orifice in a thin plate is nearly = 1, and 
therefore, so long as we have no measurements of the stream of air, 
we must put the coefficient of effiux fi = a (p equal to the coefficient 
of contraction a. The older experiments upon the efflux of air 
through orifices in a thin plate yary very considerably from each 
other. The experiments of Koch, calculated according to the 
formula for water by Buff, gave for circular orifices from 3 to 6 
lines in diameter, when the height of the water manometer was 
from 0,2 to 6,2 feet, fi — 0,60 to 0,50; on the contrary, the experi- 
ments of d'Aubuisson, calculated in the same way, give for circular 
orifices 1 to 3 centimeters in diameter, when the height of the 
water manometer is between 0,027 and 0,144 meters, fi = 0,65 to 
0,64. Poncelet also found, upon calculating the experiments of 
Pecqueur by the same formula, for an orifice 1 centimeter in diam- 
eter, under an excess of pressure of 1 atmosphere, or of a column 



§464.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 



945 



of water 10 meters high, fi = 0,563, and for a similar one 1,5 cen- 
timeters wide, [1 — 0,566. The more extended experiments of the 
author, calculated according to the last formula 

gave the following results : 

1) When the diameter of the orifice d = 1 centimeter and the 
ratio of the pressures was 



V 




1,05 


1,09 


1,43 


1,65 


1,89 


2,15 i 

1 


^- 


0,555 


0,589 


0,692 


0,724 


0,754 


0,788 



2) When the diameter of the orifice d = 2,14 centimeters, for 



1 + h 


1,05 


1,09 


1,36 


1,67 


2,01 


11 = 


0,558 


0,573 


0,634 


0,678 


0,723 



3) When the diameter of the orifice d = 1,725 centimeters, for 



b 


1,08 


1,37 


1,63 


fj, = 


0,563 


0,631 


0,665 

-■ 



4) When the diameter of the orifice d = 2 centimeters, for 



b -^h 

h 


1,08 


1,39 


fl = 


0,578 


0,641 



The coefficient of contraction for efllux through an orifice in a 
thin plate increases sensibly with the head. But if the formula for 
water is employed, there is much less variation ; this formula gives 

jE* nearly y—, e.g. for -- = 2 ; ^^ — 0,707 times as great as the 



946 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 465. 



last formula. According to the first table, ford = l and ^- = 2, 
' ' ' — = 0,771; hence, according to the water formula. 



^ = 



2 



^ = 0,707 . 0,771 — 0,555, which is nearly the same value as Pon- 
celet found. 

For efl&ux through a circular orifice 1 centimeter in diameter, 
situated in a conically convergent ivall, the angle of convergence 
being 100 degrees, the author found for 





1,31 


1,66 


\i = 


0,752 


0,793 



In like manner with the same orifice in a conically divergent 
wall, the angle of divergence being 100 degrees, the author obtained 
for 



b ~ 


1,30 


1,66 


- 


0,589 


0,663 



§ 465. The variability of the coefficient of contraction a = jj, 
for the efflux of air through an orifice in a thin plate also affects, 
according to the well-known formula 

fi = (}) = -- = — ; , (see § 422), 



"i-^ •..(!:,)• 



the coefficient of efflux for short pipes. According to the experi- 
ments of Koch, cited above, we have for such tubes 3 to 4 lines 
in diameter and from 4 to 6 times their diameter in length, when 
the pressure is 0,3 to 6,2 feet of the water manometer, fi = 0,74 to 
0,72, while, on the contrar}^ d'Aubuisson gives for similar tubes, 1 
to 3 centimeters in diameter, 3 to 4 times as long as wide, and 
under a pressure equal to 0,027 to 0,141 meters of the water ma- 
nometer, fj, = 0,92 to 0,93; and Poncelet found for cylindrical 
pipes 1 centimeter in diameter and from 2^ to 10 centimeters long, 
under twice the atmospheric pressure, fj^ = 632 to 0,650. 

The experiments made by the author, on the contrary, have led 
to the following results : 



§465.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 



947 



1) A short cylindrical tube or ajutage, 1 centimeter in diameter 
and 3 centimeters long, gave for 



h +h 
h ~ 


1,05 


1,10 


1,30 


11 = 


0,730 


0,771 


0,830 



2) A similar tube, 1,414 centimeters in diameter and three times 
as long as ivide, gave for 



h + h 


1,41 


1,69 


11 = 


0,813 


0,822 



3) A similar pipe, 2,44 centimeters luide and three times as long, 
gave for 

--y^ = l,74,f^ =: 0,833. 

The increase of the coefficient of efflux as the pressure increases 
is explained by the simultaneous increase of the coefficient of 
contraction. 

The short pipe (1), when its inlet orifice was slightly rounded 
off, gave as a mean value for its coefficient of efflux ft = 0,927, 
which is much greater than that for a similar pipe which is not 
rounded off. 

4) A short pipe, ivith its inlet orifice tvell rounded off, 1 centi- 
meter wide and 1,6 centimeters long, gave for 



I -vh 


1,24 


1,38 


1,59 


1,85 
0,971 


2,14 


fi = 


0,979 


0,986 


0,965 


0,978 



The advantage of the formula for efflux 

/ p b ' 

Qz=aFy2a~~ over the others 

•^ y p 

is shown by the fact that this coefficient approaches very nearly 
(as it should do) unity. 

The older formula gives of course for great pressures much 
smaller values for ft. 



948 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 465. 



On the contrary, the logarithmic formula (see § 460) gives much 
greater values which may sometimes even exceed unity. 

A short conical pipe, rounded off at the inlet orifice, gave nearly 
the same values for ^i, and a short conical tube, which was not 
rounded off, and which was 1 centimeter in diameter and 4 centi- 
meters long, and whose angle of convergence was 7° 9', gave for 



h + h 


1,08 


1,27 


1,65 


ix — 


0,910 


0,922 


0,964 



Koch and Buff found with a similar tube, whose exterior diam- 
eter was 2,72 lines and the angle of convergence of whose sides was 
6°, under a head of 0,3 to 6,2 feet of the water manometer /^ = 0,73 
to 0,85, and according to d'Aubuisson a similar pipe, whose orifice 
was 1,5 centimeters in diameter, gave under a pressure measured 
by a height of from 0,027 to 0,144 meters of the water manometer, 
fi zn 0,94. The old or water formula was employed in the calcu- 
lations. 

The complete nozzle A C, Fig. 736, § 434, consisting of a 

conical tube with an angle of convergence of 6°, which was 14,5 
centimeters long, 1 centimeter wide at the outlet and 3,8 centi- 
meters wide at the inlet, which was well rounded off, gave for 



1) + h 
h 


1,08 


1,45 


2,16 


fi = 


0,932 


0,960 


0,984 



By experiments upon the influx of air into vessels, Saint- 
Venant and Wantzel found for a short mouth-piece, rounded off 
internally in the form of a quarter of a circle, when the calcula- 
tions were made according to the new formula, /x — 0,98, and /or 
an orifice in a thin plate, fjb = 0,61. 

If the pressures are small, as is the case in the ordinary fan, 

where 

we employ the new formula for efflux 

Q 



< I, we can substitute, according to what precedes, when 



^ 



F \/2g^ .^ = 1299 fj, F |/(1 + 0,004 r) | cubic feet, 

as a mean 

1) for an orifice in a thin plate, fi = 0,56, 



466.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 



949 



2) for a short cylindrical pipe, fi = 0,75, 

3) for a well rounded off conical motcth-piece, fi = 0,98, 

4) for a conical pipe, whose angle of convergence is about 6°, 
fi = 0,92. 

Example. — If the sum of the areas of two conical tuyeres of a blowing 
machine is 3 square inches, the temperature in the reservoir is 15°, the 
height of the manometer in the regulator is 3 inches and the height of the 
barometer in the exterior air is 29 inches, we have the effective discharge, 
measured at the pressure of the exterior air, 



Q = 1299 fi F y(l + 0,004 r) | 



= 1299 . 0,92 . 



V 



(1 + 0,004 . 15) 



24,9 y -i 



,06 . 3 



144 y V- ' "'"--• — / 29 ■"'" '' 29 
= 24,9 . 0,331 = 8,242 cubic feet. 

§ 466. Coefficient of Friction of Air. — If air moves through 
a long pipe C F, Fig. 796, it has, like water, a resistance of friction 

Fm. 796. 
M 




to overcome, and this resistance can be measured by the height of 
a column of air, which is determined by the expression 



^=^. 



I 



V' 



d' 2/ 

in which, as in the case of water pipes, I denotes the length, d the 
diameter of the pipe, v the velocity of the air, and ^ the coefficient 
of resistance of friction, to be determined by experiment. 

Girard's experiments upon the movement of air in pipes gave a 
coefficient of resistance ^= 0,0256, those of d'Aubuisson, as a mean, 
^ = 0,0238, while according to the experiments of Buff the mean 
value of (^ = 0,0375. Poncelet, on the contrary, found from the 
data furnished by the experiments of Pecqueur, when the ratio of 

pressure is ^^ = 2, fi = 0,0237. 

The experiments of the author, calculated according to the new 
formula, gave the following results : 

1 ) A brass tube, 1 centimeter wide and 2 meters long, gave for 



950 



GENERAL PRINCIPLES OF MECHANICS. 



[§46^ 



velocities of from 25 to 150 meters ^ gradually decreasing from 
0,027260 to 0,01482. 

2) A glass tube of the same length, when the velocities were 
about the same, gave ^ = 0,02738 to 0,01390. 

3) A trass tube, 1,41 centimeters wide and 3 meters long, gave 
^ = 0,02578 to 0,01214. 

4) and a similar glass Uibe, ^ = 0,02663 to 0,009408. 

5) Finally, a zmc tube, 2,4 centimeters wide and 10 meters 
long, gave, for velocities of from 25 to 80 meters, ^ — 0,2303 tp 
0,01296. 

From what precedes we may conclude that it is only when 
velocities are about 25 meters or 80 feet, that the coefficient of re- 
sistance C can be put = 0,024, and that it becomes smaller and 
smaller as the velocity of the air in the pipe increases. 

Approximatively we can write, when the velocity is expressed 

in meters, ^ 



0,120 1 .. . ^ • ^ X r 0,217 rr, 

— -— or when it is expressed m leet ^ = — — -. The 

Vv Vv 



general relations of the flow of air in pipes are very similar to those 
of water. 

The resistance, caused by elbows and bends, is to be treated in 
the same way as in the case of water. 

In the author's experiments a rectangular elbow, 1 centimeter in 
diameter, gave ^ — 1,61, and a similar one, 1,41 centimeters in 
diameter, gave C, = 1,24, and a pipe like the former, when bent in 
the shape of a quarter of a circle, gave ^ = 0,485, and one like the 
latter, bent in the same way, gave s — 0,471. 

§ 467. Motion cf Air in Long Pipes. — By the aid of the 

coefficient ^ of the resistance of friction of a pipe B F, we can cal- 
culate the velocity of efflux and the discharge for a given length 
and width of the pipe. 

If 7^2 is the height of the manometer M.^ at the end of the pipe 
C F, Fig. 797, directly behind the mouth-piece F, whose coefficient 

Feg. 797. 




of resistance is ^ = — ^ — 1, and if d denote the diameter of the 



§ 467.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 



951 



pipe and dx that of the orifice, whose area is therefore if = -~ 
have, according to what precedes, the discharge 



we 



^9 



Q=li, F, 



Pi h. 

- == 1299 /^, TT — 



-(f 




{I + dr) 



-(I) 



7-, cub. ft. 



\d)i2g\fi,Fj' 



or, inversely, for the height h^ of the manometer 
Pi ]h _ 

But the height of the manometer at the entrance of the pipe is 

'*■==''' + ^ 5 3^' 
Z denoting the length of the pipe between M^ and M^, and v the 
velocity of the air in this pipe; hence we have 

Pi A - Ti - l^\^ A_ /_e_V 4. ^ ^ -^ or 

r^; b ~\} \dl r^gKii^Fj '^ ^'d^g'^^' 



I d\ 
substituting v = \~j)^\ and Vx ■= 



Q 



Pi Ih — 

hence the discharge is 



\1 - (I) 



\i,' ^ ^ d\d) 



2 g \fJ ' 



e = ^i 




2^ 



Pi h 



['-(l)XV'i(l)" 



= 1299 



7r6?i= 



/ 



{1 + dr) 



h 



[-(in.7-^i(i)" 



cubic feet. 



If, finally, the height h of the manometer M in the reservoir 
^ ^ is known, we have, when we denote the coefficient of resist- 
ance for the entrance 6^ by ^^ and substitute — = 1 + ^-^^ since at 



/^i 



V 



the entrance into the pipe the head C -^— is lost, 

Z g 



Pi \ ^ 
and consequently the discharge 



'■• * 4,) (I) - * <■] r, ©' 



953 GENERAL PRINCIPLES OF MECHANICS. [§ 46a 



Q-r. F, 




= 1299 ^ I / _ :__— .-^i^ cubic feet. 



If the point where the air enters the pipe is a distance s helow or 
above the point where it is discharged from it, we must subtract 

from or add to the quantity — . ^ in the numerator under the rad- 
ical sign a quantity s. 

Example. — The height of a quicksilver manometer, which is placed 
upon a regulator at the head of a system of air pipes 320 feet long and 4 
inches in diameter, is 3,1 inches, the height of the barometer in the free 
air is 29 inches, the width of orifice in the conically convergent end of the 
pipe is <ij =2 inches, and the temperature of the compressed air in the 
regulator is r = 30° C. ; what quantity of air is delivered through these 
pipes ? 

Here (1 + 0,004 r) ^ = 1,08 . ^ = 0,11545, 

^'^ = T; ~ ^ = 0;75^ - ^ =" ~9" - ^ = 9 =^ ^'^'^^' 

4 = 0,024 . 320 . 3 = 23,04, (^y= Q':- ^- = 0,0625, 

hence the required discharge is 

q = 1299 . 0,021817 |/' ^'^^^^^ 



(0,778 + 23,04) 0,0625 + 1,330 



= 28,344/ , .' ~T-^KK = 38,34 V0,040954 = 5,735 cubic feet. 
' ' 1,489 + 1.330 ' ' ' 

§ 468. Efflux when the Pressure Diminishes. — If there 
is no influx of air into a reservoir, from which an uninterrupted 
discharge of air is taking place through an orifice in it, the density 
and tension gradually diminish, and consequently the velocity of 
efflux becomes less and less. The relations of this diminution to 
the time and to the discharge can be determined in the following 
manner. 



§468.] EFFLUX OF THE AIR AND OTHER FLUIDS, ETC. 953 

- Let tlie volume of the reservoir be F, the initial height of the 
manometer be = h^, and its height at the end of a certain time t be 

= hi, and let that of the barometer in the free air be = ^ ; then 
the quantity of air originally in the reservoir, reduced to the 
pressure of the exterior air, is 

_ v(b + K) 
~ I 

and at the end of the time t it is 

b ' 

hence the discharge in the time t, reduced to the external pressure, 

is V { h + K) V {h -h AQ _ V (K - h) 

^'- 1) I - h ' 

But we have also 

X denoting the mean height of the barometer during the time t of 
efflux; hence 

Now if we put ha = m G and h^ = n o, we have the mean value 

{x)-i = ^^ {1-i +2-i +...+ m-i) - (1-^ + 2-* + . . . + n-^A 
m—n ^ ' 

m—n\^ ^ I m — n V g ^ a) 

= —^7 — ^, — — ^ — :r- — 1 -(seelngenieur,p. 88); 

(m — n) G . h^ — 111 ° ^r /> 

iience the required time of efflux is 

_ ^V(V\- V X) _ 2F i^/K JTa 

^ ~ 7="^"^ ~ / — ^ V b ~^ Tr 

fiFy2ff-^-b fj,Fy2ff^~ 

This determination is sufficiently correct only when the reser- 
voir (F) is large, or when the orifice of efflux, as well as the 
pressure, is small, in which case the cooling of the air in the reser- 
voir is very slight. 

Example. — A cylindrical regulator 50 feet long and 5 feet in diameter is 
filled with air at a pressure corresponding to the height h = 10 inches of 
the manometer and at a temperature of 6° C. Now if the air issues from 
an orifice 1 inch in diameter into a space where the barometer stands at 
27 inches, the question arises, in what time will the manometer sink to 7 



954 GENERAL PRINCIPLES OF MECHANICS. [§468. 

inches and what will be the discharge in that time ? The volume of tho 
regulator or boiler is 

F = T . 5' . 50 = 1350 . J = 981,75 cubic feet, and 

4 4 ' ' 






2 r/-?i = 1399 Vl + 0,00367 . r = 1399 Vl,03303 = 1313 and 

F ^"^ (yV)' = ^^f. = 0,005454 square feet. 

Now if we put the coefficient of efflux // = 0,95, we have the required 
duration of the efflux < 

3.981,75.0,09942 
' = ¥,95rO-0054M-.l31'S = ^^'^^ ^^^^^^^- 

Remark. — A more general theory of the efflux of air and steam will be 
given in the second volume. 

Final Remark. — Experiments upon the efflux of air have been made 
by Young, Schmidt, Lagerhjelm, Koch, d'Aubuisson, Buff, and more re- 
cently by Saint Tenant, Wantzel, and Pecqueur. In reference to the ex- 
periments of Young and Schmidt, see Gilbert's Annalen, Vol. 22, 1801, and 
Yol. 6, 1820, and Poggendorfs Annalen, Yol. 2, 1834 ; for those of Koch 
and Buff, see the " Studien des Gotting'schen Yereines bergmannischer 
Freunde," Yol. 1, 1824; Yol. 3, 1833; Yol. 4, 1837; and Yol. 5, 1838 ; 
also Poggendorfs Annalen, Yol. 27, 1836, and Yol. 40, 1837. See also 
Gerstner's " Mechanik," Yof. 3, and Hiilsse's "Algemeine Maschinenency- 
klopadie," Article " Ausfluss." Lagerhjelm's experiments are discussed in 
the Swedish work " Hydrauliska Forsok af Lagerhjelm, Forselles och 
Kallstenius," 1 Delen, Stockholm, 1818. The experiments of d'Aubuisson 
are to be found in the " Annales des Mines," Yol. 11, 1825 ; Yol. 13, 1826 ; 
Yols. 3 and 4, 1828 ; and also in d'Aubuisson's " Traite d'Hydraulique." 
The experiments of Saint-Yenant and Wantzel are to be found in the 
" Comptes rendus hebd. des seances de 1' Academic des Sciences, a Paris, 
1839." The latest French experiments are discussed by Poncelet in a 
"note sur les experiences de M. Pecqueur relatives a I'ecoulement de I'air 
dans les tubes, etc.," which is contained in the Comptes rendus, and an 
abstract of it is to be found in the Polytechnische Centralblatt, Yol. 6, 
1845. From these experiments Poncelet concludes that air follows the 
same laws of efflux as water. The greater number of these experiments 
were made with very narrow orifices, for which reason they scarcely fulfill 
the requirements of practice. Unfortunately these experiments do not 
agree as well as could be wished, and the coefficients found by d'Aubuisson 
differ very sensibly from those calculated from Koch's experiments. Com- 
parative experiments upon the efflux and influx of air and upon the efflux 
of water are given in the author's "Experimental-Hydraulik." The re- 
sults of the latest experiments of the author, which were made upon a 
large scale, are given in the 5th volume of the Civilingenieur. 



§ 469.] THE MOTION OF WATER IN CANALS AND RIVERS. 955 

CHAPTER VII. 

OF THE MOTION OF WATER IN CANALS AND RIVERS. 

§ 469. Running Water. — The tlieory of the motion of water 
in canals and rivers forms the second part of hydrauhcs. Water 
flows either in a natural or in an artificial ied (Fr. lit ; Ger. Bett). 
In the first case the channel is a river, creek, rivulet, etc., in the 
second case it is a canal, ditch, race, trough, etc. In the theory of 
the motion of running water this difference is of but little im- 
portance. 

The heel of the stream consists of the 'bottom of the channel 
(Fr. font du lit; Ger. Grundbett or Sohle) and of the two hanhs 
or shores (Fr. bords ; Ger. Ufer). If we pass a plane through the 
stream of water at right angles to the direction, in which it is 
flowing, we obtain a transverse section (Fr. section ; Ger. Quer- 
schnitt). The line bounding this section is the tranverse profile 
v\'hich is composed of the tvater profile or tvettecl p)erimeter and of 
the air p)rofile. A vertical plane in the direction of the stream 
gives us the longitudinal section or profile (Fr. profil ; Ger. Pro ill) 
of the latter. The slojoe of the stream (Fr. pente ; Ger. Abhang) is 
the angle formed by its surface with the horizon. The relative 
slope is the fall in the unit of distance. The slope is determined 

for any definite distance by ih^fall (Fr. 
Fig. 798. chute ; Ger. Gefalle), which is the vertical 

distance of one of the extremities of a cer- 
tain portion of the stream above the other. 
In the portion A D = I, Fig. 798, B C is 
the bottom of the channel, D H = h the 
fall and the angle D A IT = d is the slope. The relative slope is 

sin. d = p or approximatively 6 = -. 

Remark. — The fall of creeks and rivers varies very mucli. The Elb 
falls in a German mile (4|^ English miles) from Hohenelbe to Podiebrad 
57 feet, from there to Leitmeritz 9 feet, from there to Miihlberg 2,5 feet. 
Mountain streams fall from 8 to 80 feet per mile. For particulars see 
" Vergleichende hydrographische Tabellen, etc., von Stranz." The fall in 
canals and other artificial channels is much smaller. The relative slope is 
generally less than 0,001, it is often 0,0001 and even less. More details 
upon this subject will be found in the second part. 




956 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 470- 



§ 470. Different Velocities in a Cross -section. — The ve- 
locity of the water is far from being uniform in all points of the 
same transverse section. The adhesion of the water to the bed of 
the channel and the cohesion of the molecules of water cause the 
particles of water nearest to the sides and bed of the channel to be 
most hindered in their motion. For this reason, the velocity 
decreases from the surface towards the bed of the channel and it is 
a minimum at the shores and bottom. The maximum velocity in 
a straight river is generally found in the middle or in that portion 
of the surface, where the water is the deepest. That portion of the 
river, where the water has its maximum velocity, is called the line 
of current or axis of the stream and the deepest portion of the bed 
is called the mid-channel. 

When the stream bends, the axis of the stream is general near 
the concave shore. 

The mean velocity of the water in a cross-section, according to 
§ 396, is 

__ Q __ Discharge per second 

~ F ~ Area of the transverse section* 

We can also determine the mean velocity from velocities Ci, Cj, Ca? 
etc., in the different portions of the transverse section and the 
areas F^, F^, F^, etc., of the latter. We have here 

and, therefore, also 

Fx + -if^2 + • • • 

Besides the mean velocity we introduce the mean depth of water, 
I.E., that depth a, which a transverse section would have, if its 
area was the same and the depth was uniform instead of being 
variable and equal to «i, a^, a^, etc. Here we have 

_ F _ Area of the transverse section 
h Width of the transverse section* 
If the widths of the elements corresponding to the depths a^, «2> 

«3, etc.. Fig. 799, are &i, l^, h, etc., we 
have 

F — axhx + 6?2 ^2 + • • •> 
and consequently 

_ fl^l 5l + «2 ^2 + • • . 

a — '^ ^ ^» 

^1 4- ^2 + . . . 

Finally, the mean velocity is 



Fig. 799. 




§471.] THE MOTION OF WATER IN CANALS AND RIVERS. 957 

^1 ^1 ^1 + «2 ^2 ^2 + . . . 



Q 3^ 

^1 ^1 + «2 ^2 + • . . ' 

and, when the widths ^1, b<i, etc., of the portions are the same, 

Cl\ C\ ~H t?2 ^2 ~T" • • • 

C = . 

• Ci\ "T~ ^2 ~i~ • • • 

A river or creek is in a state of permanency (Fr. permanence ; 
Ger. Beharrungszustande) or it has a fixed regimen, when the same 
quantity of water passes through each of its cross-section in the 
same time, i.e., if Q or the product F c of the area of the cross- 
section and the mean velocity is constant for the whole length of 
the portion of the river under consideration. Hence we have the 
simple law : ivlien the motion of the water is permanent the mean 
velocities of tivo transverse sections are to each other inversely as the 
areas of these sections. 

Example — 1) In the transverse section A B G D^ Fig. 799, of a canal, 
■we have found the widths of the divisions to be 

5j z= 3,1 feet, &g = 5,4 feet, Jg = 4,3 feet, 
the mean depths to be 

a^ = 3,5 feet, a^ = 4,5 feet, a^ = 3,0 feet 
and the corresponding mean velocities to be 

Cj = 3,9 feet, Cg = 3,7 feet, e^ = 3,3 feet. 
Here we can put the area of the section 

i^' = 3,1 . 3,5 + 5,4 . 4,5 + 4,3 . 3,0 = 44,95 square feet 
and the discharge 

^ = 3,1 . 3,5 . 3,9 + 5,4 . 4,5 . 3,7 + 4,3 .3,0 . 3,3 = 153,665 cubic feet, 
from which we obtain the mean velocity 

^^_153^65_ 
'^- F - 44,95 ~ "^'^^^ ^^^^• 
3) K a ditch should carry 4,5 cubic feet of water with a mean velocity of 

. 4,5 

2 feet per second, we must make the area of its transverse section -^ = 3,35 

square feet. 

3) If the same river is at one place 560 feet wide and as an average 9 
feet deep, and if it moves with a mean velocity of 2^ feet, the mean velocity 
at another place, where it is 330 feet wide and as a mean 7,5 feet deep, is 

560 . 9 
' = 32077-5- 2-^5= *'^^^^«'='- 

§ 471. Mean Velocity.— If we divide the depth of the water 
at any point into equal parts and lay off the corresponding veloci- 
ties as ordinates, we obtain a scale A B, Fig 800, of the velocities 
of the stream. Although it is very probable that the law of this 
Bcale, or of the change of velocity, is expressed by a curve, as 




958 GENERAL PRINCIPLES OF MECHANICS. [§471.; 

E. G. according to Gerstner, by an ellipse, etc., yet without risking 

a very great error we can substitute a 
Fig. 800. straight line, i.e., assume that the velocity 

^Co diminishes regularly with the depth ; for 
n-j^ this diminution of the velocity is always 
slight. According to the experiments of 
Ximenes, Briinnings and Funk, the mean 
velocity in a perpendicular line is 
Cm ^^ 0,915 Cq, 
c^ denoting the maximum velocity or that of the surface of the 
water. The diminution of the velocity from the surface to the 
middle if is therefore 

c,-Cm = (l- 0,915) c, = 0,085 c,, 
and we can put the velocity at the bottom, or at the foot of the 
perpendicular, 

c^ = c, - 2. 0,085 c,= {1 - 0,170) c, = 0,83 c,. 
If the total depth is a, we have, if we assume the scale of velocity 
to be represented by a straight line, for a depth A JS/' = x below 
the water the velocity 

V z= c, - {c, ~ c„) ^= (l ^ 0,17 ^ c,. 

Now if Co, Ci, Cg are the velocities at the surface of a profile, 
whose depth is not very variable, we have the corresponding veloci- 
ties at the mean depth 

0,915 c„, 0,915 Ci, 0,915 c„ 
and therefore the mean velocity in the whole transverse section 

A A-1 K ^0 "t" ^1 ~^~ ^2 + . . . + Cn 

= 0,915 z . 

n -h 1 

If, finally, we assume that the velocity diminishes from the line 
of current or axis of the stream towards the shores in the same 
manner as towards the bottom, we can put the mean superficial 
velocity Cp + Ci -\- ..• + <?._ ^ q. ^ ^ . 
^m ~ ^'^^^ "^^ ' 

thus we obtain the mean velocity of the whole transverse section. 

c = 0,915 . 0,915 . c, = 0,837 c,, 
I.E.J 83 to 84 per cent, of the maximum velocity. 

Prony deduced from the experiments of du Buat, which, how- 
ever, were made in small ditches, the following formula, which is 
perhaps more correct in such cases, 

/2,872 4- cA , / 7,78 -^ cA . . 

'- = (3A53-T^) '» '^'^' = {mrr^J '" ^''^' 



§ 472.] THE MOTION OF WATER IN CANALS AND RIVERS. 959 

Hence for mean velocities of 3 feet we have 

c^ = 0,81 Co. 

If the flow of the water is impeded by a contraction of the 
transverse section, the level of the water will be raised, and c^ be- 
comes sdll greater. 

Example. — If the velocity of the water in the axis of a river is 4 feet, 
and if its depth 6 feet, we have the mean velocity in the corresponding 
perpendicular 

Cr, = 0,915 . 4 = 3,66 feet, 
the velocity at the bottom 

= 0,83 . 4 = 3,32 feet, 
the velocity 2 feet from the surface 

V = (1 - 0,17 . I) 4 = (1 - 0,057). 4 = 3,772 feet 
and, finally, the mean velocity of the entire transverse section 

c ^ 0,837 . 4 = 3,348 feet ; 
on the contrary, according to Prony, w^e would have 

11,78 , 23,56 
' = Wi-^=-7jY = ^^''''''- . 
Remakk. — This and the following subjects are treated at length in the 
Allgemeine Maschinenencyklopildie, Article " Bewegung des Wassers." 
New experiments and new views upon the same subject are to be found in 
the following work : " Lahmeyer, Erfahrungsresultate uber die Bewegung 
des Wassers in Flussbetten und Canalen, Braunschweig, 1845." Accord- 
ing to Baumgarten's observations (see Annales des Fonts et Chaussees, 
Paris, 1848, and also polytechnisches Centralblatt, No. 14, 1849) the values 
given by this formula, when the velocities are great (above 1,5 meters), are 
too large and we must put in such cases 

/2,372 + cA ^^ 

'- = \sj5sTvJ ' ^'^ '^ ^"*""'- 

Owing to the resistance of the air the maximum velocity of the water is 
to be found a little below the surface of the water. From this point of 
maximum velocity the velocity diminishes as the square of the depth ; hence 
the scale of velocity corresponds to a parabola. In like manner, according 
to Boileau (see his Traite sur la mesure des eaux), the velocity decreases 
as the square of the distance from the axis of the stream. If c^ denotes the 
velocity in the axis of the stream, the velocity at the horizontal distance x 
from it will be 

c, = Co — ^ aj% 
in which /i denotes an empirical number, which is different for different 
streams. 

§ 472. Most Advantageous Transverse Profile. — The 

resistance, offered by the bed of the stream in consequence of the 
adhesion, viscosity and friction of the water, is proportional to 
the surface of contact, and consequently to the wetted perimeter 



960 GENERAL PRINCIPLES OF MECHANICS. [§473. 

f, or to that portion of the profile which forms the bed. Now since 
the number of filaments of water passed by any transverse section 
increases with its area, the resistance to each filament is inversely 

proportional to the area, and consequently to the quotient ^ of the 

wetted perimeter divided by the area F of the entire transverse 
section. In order to have the least resistance from friction, we 

must give the profile such a shape that -^ shall be as small as pos- 
sible, I.E., that the wetted perimeter jp shall be a minimum for a 
given area, or that the area shall be a maximum for a given wetted 
perimeter p. When the apparatus which conducts the water is 
closed on all sides as in the case of pipes, p is the perimeter of the 
entire transverse section. N'ow among all figures of the same 
number of sides, the regular one, and among all the regular ones, 
the one with the greatest number of sides has the smallest perim- 
eter for a given area; hence in conduits closed on all sides the 
resistance is smaller the more regular the shape of their transverse 
section is, and the greater the number of sides is. Since the circle 
is a regular figure of infinite number of sides, the resistance of 
friction is the smallest when the transverse section is of that form. 
When the aqueduct is open on top, the case is different ; for the 
upper surface is free, or in contact with the air alone, which, so 
long as it is still, offers little or no resistance to the water. We 
must, therefore, in determining this resistance of friction, neglect 
the air profile. 

In practice we employ in canals, ditches, troughs and flumes 
only rectangular and trapezoidal profiles. A horizontal line E F, 
Fig. 801, passing through the centre M of the square A C, divides 

the area and perimeter into two equal parts, and 
what has been said of the square is true for these 
halves; hence, among all rectangular profiles, 
the half square A F, or that which is twice as 
vjide as high, is the one which causes the smallest 
resistance of friction. 

In like manner, the regular hexagon ACE, 
Fig. 802, is divided by a horizontal line C F into two equal trape- 
zoids, each of which, like the entire hexagon, has the greatest 
relative area, and consequently among all trapezoidal profiles, the 
half of the regular hexagon, or the trapezoid A B O F, with the 




473.J THE MOTION OF WATER IN CANALS AND RIVERS. 961 



angle of slope B C M ^=^ 60°, is the one which causes the least 
resistance of friction. 

In like manner, tlie half of a regular octagon A D E, Fig. 803, 
the half of a regular decagon, etc., and finally the half circle ABB, 
Fig. 804, are, under the proper circumstances, the most advan- 



FiG. 802. 



Fia. 803. 





Fig. 804. 




tageous profiles for canals, etc. The trapezoidal, or half hexagonal, 
cross-section causes less resistance than the half square or rec- 
tangle, the ratio of whose sides is 1 : 2 ; the relative perimeter of 
the hexagon is smaller than that of the square. The half decagon 
offers still less resistance, and with the semicircle the latter is a 
minimum. The circular and square profiles are employed only 
for troughs made of iron, stone, or wood. The trapezoid is em- 
ployed in canals, which are dug out or walled up. Other forms 
are rarely used, owing to the difficulty of constructing them. 

§ 473. When canals are not walled up, but only dug in the 
earth or sand, an angle of slope of 60° is too great or the relative 
slope cotg. 60° = 0,57735 too small; for the banks would not be 
sufficiently stable; we are therefore compelled to employ trapezoi- 
dal transverse profiles, in which the inclination of the side to the 
base is smaller than 60°, perhaps only 45° or even less. For a trapezoi- 
dal cross-section A B C B, Fig. 805, which has the same area and 
perimeter as the half square, the relative slope is = |, and the 
angle of slope is 36° 52'. If we divide the height B E into three 
equal parts, the bottom B C is equal to two of them, the parallel 
top A B is equal to 10 and each side A B = C B is = 5 parts. 
In many cases we make the relative slope = 2 ; in which case the 
angle is 26° 34', and sometimes it exceeds even 2. 

In any case the angle of slope B A E = 6, Fig. 806, or the slope 

A E 

P = cotang. 6 is to be considered as a given quantity, dependent 
B E 

upon the nature of the ground in which the canal is excavated, 

and therefore we have only to determine the dimensions of the pro- 

61 



962 GENERAL PRINCIPLES OP MECHANICS. [§473. 

file which will offer the least resistance. Putting the width B C of 



Fig. 805. 



Fig. 806. 





A E 
the bottom = b, the depth B E — a and the slope -^—p ~ ^? we 

have the wetted perimeter of the profile p = 

• AB + BC+CI) = b-{-2 Va^^~a' = b + 2 aVl + v\ 

and the area of the same 

F~ab-]-vaa = a{b-\-va), 
or inversely 

= V a, 



a 



whence the ratio 



F a F ^ ^ 

Substituting instead of a, a + x, in which i^ is a small quan- 
tity, we have 

F a -^ X 

a -\- X 
~F 



1 /^ X x\ 
a \ a a I 



+ ('^ (3 /.-nn 



V) 



+ --^ (3 Vv^ + 1 - v) 



a 



V) +( 



2 S^V + 1 



v^ 1_\ 



X + 



ic' 



a ^ \ J^ a / a 

In order that this value, not only for a positive but also for a 
negative value of x, shall be greater than the first value 

I + ^^ (2 vv^n - v), 

or that ^ shall be a minimum, it is necessary that the members 
with the factor x shall disappear or that 

whence the required depth of the canal is 

F 



a"" 



2 |/i^' + 1 



V 



or, since v — cotang. 6 and Vv^ -}- 1 = 



sin. 6^ 



§474.] THE MOTION OF WATER IN CANALS AND RIVERS. 963 

F sin. 6 



a' 



2 - COS. e: ; 

Hence for a given angle of slope d and for a given area, the 
most advantageous for m for the transverse profile is determined by 
the formulas 



- i/ZL!^5 ^ ^ ' ^ 



a = y ^ and b = a cotang, 6. 

4i — COS. tf a 

Consequently the width A D of the top is 

F 
hi = I)-\-2va = [-a cotang. d, 

and the ratio 

iL - A _?JL_ - 1 (2 - COS. 6) a _ 2 
F ~ F '^ F siTiTd ~ a'^ ~ F sin. d ~ a 
Example. — What dimensions should be given to tlie transverse profile 
of a canal, when the angle of slope of its banks is to be 40° and when it is 
to carry a quantity ^ = 75 cubic feet of water with a mean velocity of 
3 feet. 
Here 

75 

-^ = — = -^ = 25 square feet, and therefore the required depth is 



, / 25 sin. 40° ^ ,/ 0,64279 

the width at the bottom is 

25 

I =-■ 3-609 ~ ^'^^^ cotang. 40° = 6,927 - 4,301 = 2,626 feet, 

the horizontal projection of the slope of the shore is 

V a = a cotang. = 3,609 cotang. 40° = 4,301, 

the width on top is 

h^ =b + 2a cotang. d = 6,927 + 4,301 = 11,228 feet, 

the wetted j)erimeter is 

2/7 7 218 

p = h + 4—^ = 2,626 + -^^o- = 13,855 feet, 
■^ sin. 6 ' sin. 40 

and the ratio which determines the resistance of friction is 

P -^- ^ - 5542 
.P - « - 3,609 - ^'^^^^' 

We have for a transverse profile in the shape of the half of a regular 

hexagon, where d = 60°, a = 3,80 feet, h = 4,39, h^ = 8,78 andp ^ 18,16 

feet, and therefore p 13,16 ^ ^^„ 

§ 474. Table of the Most Advantageous Transverse 
Profiles. — The following table gives the dimensions of the most 
advantageous transverse profiles for different angles of slope and for 
given transverse sections : 



964 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 474. 



Angle of 
slope 9. 



90" 
60° 

45° 
40° 

36° 52 

35° 

30° 

26° 34' 

Semi- 
circle 



Relative 
slope V. 



0,577 



1,000 



1,192 

1,333 
1,402 

1,732 

2,000 



DIMENSIONS OF THE TRANSVERSE PROFILES. 



Deptl 


1 a. 


0,707 


Vf 


0,760 


Vf 


0,740 


Vf 


0,722 


Vf 


0,707 


Vf 


0,697 


Vf 


0,664 


Vf 


0,636 


Vf 


0,798 


Vf 



Width of bot- 
tom d. 


1,414 


Vf 


0,877 


Vf 


0,613 


Vf 


0,525 


Vf 


0,471 


Vf 


0,439 


Vf 


0,356 


Vf 


0,300 


Vf 




- 



Horizontal pro- 
jection of slope 
v a. 







0,439 


Vf 


0,740 


Vf 


0,860 


Vf 


0,943 


Vf 


0,995 


Vf 


1,150 


Vf 


1,272 


Vf 







Width at the 
top ^-{- 2 V a. 



1,414 Vf 

1,755 ^^ 
2,092 Vf 

2,246 Vf 

2,357 ^^ 
2,430 Vf 

2,656 Vf 

2,844 Vf 

1,596 Vf 



Quotient 



F 



Y F 



2,828 

'V~F 
2,632 

V~F 
2,704 

V~F 
2,771 

Vf 
2,828 

Vf 
2,870 

sHf 
3,012 



F 

3,144 



F 

2 ,507 

'Vf 



p 



and = 



We see from the above table that the quotient ^7- is a minimum 
for the semicircle, that it is greater for the half 



Vf 



hexagon and still greater for the half square, and for the trapezoid 
with its sides sloping at an angle of 36° 52', etc. 

Example. — What dimensions are to be given to a transverse profile 
whose area is to be 40 feet, when the banks are to slope at an angle of 35 "" 
According to the foregoing table 

the depth is a = 0,697 Vio = 4,408 feet, 

the lower breadth is 5 = 0,439 V40 = 2,777 feet, 

the horiaontal projection of the slope v a=. 0,995 ViO = 6,293 feet, 
the upper breadth l^ = 15,363, 



§ 475.J THE MOTION OF WATER IN CANALS AND RIVERS. 965 

and the quotient is 

f = ?^ = 0,4538. 

§ 475. Uniform Motion. — The motion of water in channels 
is for a certain distance either uniform or variable ; it is uniform, 
when the mean velocity in all the cross-sections is constant, and, on 
on the contrary, it is variable, when the mean velocity and also the 
area of the cross-sections change. We will now treat of uniform 
motion. 

When the motion of water is uniform for a distance A D = I, 
Fig. 807, the entire fall li is employed in overcoming the friction 

upon the bed, and the water flows away 
with the same velocity, with which it 
arrived, i.e., a height due to a velocity is 
neither absorbed nor set free. If we meas- 
ure this friction by the height of a column 
of water, we can put the latter equal to 
the fall. The height due to the resistance of friction increases 

with the quotient ^, with I and with the square of the mean ve- 
locity c (§ 427) ; hence the formula 

■ 1)^ = ^^-^ 

holds good, in which ^ is an empirical number, which is called the 
'coefficient of the resistance of friction. 
By inversion we have 




= /.^ 



To determine the fall from the length, the transverse profile 
and the velocity, or inversely, to determine the velocity from the 
fall, the length and the transverse profile, it is necessary to know 
the coefficient of friction ^. According to Eytelwein's calculation 
of the 91 experiments of du Buat, Briinings, Funk and Woltmann, 
^ = 0,007565, and therefore 

h z= 0,007565 . -^ . ^. 
F 2 ff 

If we put g = 9,809 meters or 32,2 feet, we obtain for the 

metrical system 

h = 0,0003856 ^-^ . c' and c = 50,9 i/— y, 
F pi 



966 GENERAL PRINCIPLES OF MECHANICS. [§ 476. 

and for the English system of measure 

h = 0,00011747 -^ c' and c = 92,26 j/ ^4- 

For conduit pipes -^ = -, y^ = -^ ; hence the formula for 

pipes is 

A = 0,03036 l.f^, 

while we found more correctly (§ 428) for medium velocities in 
the same 

7i = 0,025 ~ . ^c-. 
d 2 g 

The friction upon river beds is, therefore, as might be expected, 

greater than in metal conduit pipes. 

Example — 1) How much fall must a canal, whose length is Z = 2600 
feet, whose lower width is 5 = 3 feet, whose upper width is b^ =7 feet 
and whose depth is a = 3 feet, have in order to carry 40 cubic feet of 
water per second ? Here 

p = S + 2 V2V+ 3^ = 10,211, F = \ - = 15 and c = fo = f ; 

hence the required fall is 

^ 2600 . 10,211 ,„,, 0,305422 . 10,211 .64 , ,^ , 
h = 0,00011747 . ^^^ qy = ^^—^ = 1,48 feet. 

2) What quantity of water will be delivered by a canal 5800 feet long, 
when the fall is 3 feet, its depth 5 feet, its lower breadth 4 feet and its 
upper breadth 12 feet ? Here 

P - 4 4- 2 V5^n- _ 16,806 _ 
Y - 5". 8 - ~^0 - "'^^^"^^ ' 

hence the velocity is 

c = 92,26 



/ 3 _ 92,26 _ 92,26 

^ 0,42015 . 5800 ~ V0,14005 . 5800 ~ V812;29 

= IJ = a,m feet, 

and the quantity delivered is 

Q = Fc= 4:0 . 3,237 = 129,48 cubic feet. 

§ 476. Coefficients of Friction. — The coefficient of friction, 
for which we assumed in the foregoing paragraph the mean value 
0,007565, is not constant for rivers, creeks, etc., but, as in the case 
of pipes, increases slightly, when the velocity diminishes, and 
decreases, when the velocity increases. We must therefore put 

or to some similar formula 



I 476.] THE MOTION OF WATER IN CANALS AND RIVERS. 967 

The author in the article quoted in the remark of § 471 found 
from 255 experiments, most of which were made by himself, for 
English measures 

? = 0,007409 (l + A^\ 

and for the metrical system of measures 

^= 0,007409 (l/^^). 

We see that this formula gives for a velocity c = 8| feet the 
above-quoted mean value C = 0,007565. In order to facilitate cal- 
culation, the following tables for the metrical system have been 
prepared : 



Velocity c = 


0,1 
1175 


0,2 

0958 


0,3 

0885 


0,4 
•0849 


0,5 

0828 


0,6 
0813 


0,7 
0803 


0,8 
0795 


0,9 

0789 


meters. 


Coefficient of re- 
si stance f = 0,0 





Velocity c = 


1 


1,2 


1,5 


2 


3 


4 


5 meters. 


Coefficient of resistance 

c=o,o 


0784 


0777 


0771 


0763 


0755 


0752 


0750 



For English system of measures we can employ the following 
table. 



Velocity c = 


0,3 


0,4 


0,5 


0,6 


0,7 
0944 


0,8 
0918 


0,9 


1 


n 


2 


3 

0788 


5 
0769 


7 
0761 


10 
07551 


15 feet. 


Coefllcient of re- 
sistance ^ = 0,0 


1215 


1097 


1025 


0978 


0899 


0883 


0836 


0812 


07504 



These tables are directly applicable to all cases, where the velo- 
city c is given and the fall is required, and when formula No. 1 of 
the foregoing paragraph is employed. If the velocity c is unknown, 
or if that is the required quantity, the tables can only be employed 
directly when we have an approximate value of c. The simplest 
v>'iiy to proceed is to determine c approximatively by one of the 
formulas 



/W Ji /~Fli 

c = 50,9 V —r meters or c =^ 92, 26 i/ — y- feet, 
^ pi ' ^ pi 

then find ^ by means of the table, and substitute the value so found 

in the formula 



968 GENERAL PRINCIPLES OP MECHANICS. [§476. 



/ F 

From the velocity c we determine the quantity of water Q = F c» 

If the quantity of water and the fall are given, as is often the 

case in laying out canals, and it is required to determine the trans- 

7) 771 

yerse profile, we must substitute ~ = — ^1=: (see table, § 474) and 

c = ^ in the formula 

h = 0,007565 -^ . -^, and put 
' F % g' ^ 

771 I Q^ 

Ji = 0,007565 T7 — j^—, from which we obtain 

z g I* "^ 

F = (o,007565 ^^^'V, I.E. in meters 
\ ^ g fi J 

F = 0,043] { ^l^y , and in English feet; 

^^0,0268(^)1 
From this we obtain the approximative value 

Q 

^ ~ F' 
if we take the corresponding value of ^ from the table, we can cal- 
culate more accurately 

from which we deduce more correct values for 

Q 

c = -~r and p = m VF, 

as well as for a, i, etc. 

. Example — 1) What must be the fall of a canal 1500 feet long, whose 

lower breadth is two feet, whose upper breadth is 8 feet, and whose depth 

is 4 feet, when it is required to convey 70 cubic feet of water per second ? 

Here 

jp = 3 + 3 V4' + 3^ = 12, i?^ = 5 . 4 = 20, c = f ^ = 3,5 ; 
hence 

^ = 0,00784 and 

^.^«o. 1500 . 12 8,52 86,436 

h = 0,00784 ^ 'h = -^ = 1,34 feet. 

' 20 2g 2 g ' 



§ 477.J THE MOTION OF WATER IN CANALS AND RIVERS. 969 

3) What quantity of water is carried by a creek 40 feet wide, whose 
mean depth is 4|^ feet, and whose wetted perimeter is 46 feet, when it falls 
10 inches in 750 feet ? Here we have approximatively 



. = 92,26 . |/ ,-^gV:^ = vll~ - ^'^ '''' ' 



V, 



hence we can assmne C = 0,00765. 

We have now more correctly 

e" _ Fh _^ 4^.40.10 _ 1 _ _ 

27 ^ TT7 ~ 0,00765 . 46 . 750 ."12 ~ 1/7595 = ^'^^^^ ^"""^ "" ~ ^'^^ ^^^*' 
The quantity of water carried is 

^ = i^c = 4,5 . 40 . 6,05 = 1089 cubic feet. 
3) It is required to excavate a ditch 3650 feet long, which, with a total 
fall of one foot, shall carry 12 cubic feet of water per second. What must 
be the dimensions of the transverse section when the form is a regular hex- 
agon ? Here m = 2,632 (see table, § 474) ; hence we have approximatively 
F = 0,0268 (2,632 . 3650 . 144)1 = 7,66 square feet, and 

13 , 

^ = 7:66 = ^'^^^- 

Here we must take C = 0,0083, and, therefore, 

„ / 3650 . 144\f 

Fz= ^0,0083 . 2,632 . — — ^ — \ = 7,95 square feet. 

From this we obtain the depth 

a = 0,760 VF = 2,14, the lower width 

d = 0,877 VF= 2,47, and the upper width 
\=2 . 2,47 = 4,94. 
Remake — 1) According to Saint Tenant, we can put accurately eneugh 

h = 0,000401 |r . vn = 0,000401 . 2 g . v-t\ .^ .-^ meters ; 
^ F ^ F 2 g 

hence the coefficient of resistance is 

C = 0,000401 .2g. v-t\ = 0,007887 «-tV, 
E.G. for V = 1 meter 

C = 0,007887 
and for v = ^ meter 

C = 0,007887 'V4 = 0,007887 . 1,134 = 0,008945. 
(Compare § 428, Remark 3.) 

2) A table, which abridges these calculations, is given in the Ingenieur, 
pages 460 and 461. 

§ 477. Variable Motion. — The theory of the variable motion 
of water in channels can be referred to the theory of uniform mo- 
tion, when we consider the resistance of friction upon a small por- 
tion of the length of the river to be constant and put the corre- 
sponding head 



970 GENERAL PRINCIPLES OF MECHANICS. [§477. 

We must also take into consideration the vis viva corresponding to 
the change of velocity. 

Let A B C D, Fig. 808, be a short portion of the channel of a 
river, whose length A D — I, and whose fall D H ^ Ji', let v^ be 
the velocity of approach and v^ that with which the water flows 

away. If we apply the laws of efflux to 
Fia. 808. ^j^ element D at the surface of the 

^ .^ water, we have for its velocity Vi 



D V,' , V,' 

^ but an element E, which is situated 

below the water, has on one side, it is 
true, a greater head A G = E H, but it is pressed back by the 
water below it with a head D E\ hence the effective fall, which 
produces motion, is only D II — EH— ED, and consequently 
the following formula holds good for any element : 

if we add the resistance of friction, we obtain 

in which p, F and v denote the mean values of the wetted perim- 
eter, the transverse section and the velocity. If F^ denote the area 
of the upper and F^ that of the lower transverse section, we can put 

W 4- W 

F = -^— — ? and Q = F^ v, = F^ v,, whence 

V,' - V,' _ 1 r/Q_y _ /<2 \'l / 1 _ M _£^ ^ 
2g ~ 2g 1\fJ [fJ J " \F: F,') 2 / ^"""^ 

F K^F, \f: ^ F,V F, + F^ 
from which we obtain 

1) ^ = [:^, - :^ + ? -^^r^ (^ + ^)] ^, and 

By the aid of formula 1) we can calculate from the quantity of 
water carried, the length and transverse sections of a section of river 



§ 477 ] THE MOTION OF WATER IN CANALS AND RIVERS 971 

or canal the corresponding fally^, and by the aid of formula 2) from 
the fall, length and cross-section the quantity of water carried. In 
order to obtain greater accuracy we should calculate these for sev- 
eral small portions of the channel of the river and then take the 
arithmetical mean of the results. If the total fall only is known, 
we must substitute this value instead of li in the last formula and 

instead of 

J^ _ JL^ J^ 1^ 

-'^ 1 -*^ n 

in which F^ denotes the area of the last cross-section, and instead of 



^. 



I jp 



F\ \F^' ^ F^r 



the sum of all the similar values for the different portions of the 
channel of the river. 

Example.— A creek falls 9,6 inches in 300 feet, the mean value of its 
wetted perimeter is 40 feet, the area of its upper transverse section is 70 
square feet, and that of its lower is 60 square feet. Required the discharge 
of this brook. Here 

8,025 VO^ 



q = 



i/ 1 1 AAA^r^^ 300.40/1 1\ 

4/ — — h 0,007565 . I 1 -\ 

y 602 703 ^ ' 130 \602 ^ 70V 

^^™ ^'^'^:i.=.. = 354^ cubic feet. 



V0,0000731 + 0,0003365 V0,0004096 

2 709 

The mean velocity is — — ^— - = —— = 5,45 feet ; hence it is more cor- 

Fq + F^ 130 

rect to put C = 0,00768 instead of 0,007565, 

and therefore we have more accurately 

Q = — ~- = 352,5. 

V0,0000731 + 0,0003416 
If the same stream has at another place the same amount of water in it 
and falls 11 inches in 450 feet, and if the area of its upper transverse section 
is 50 and that of its lower 60 feet, the mean length of its wetted perimeter 
being 36 feet, we have 

^ _ 8,025 Vo79l67" 



/I 1 450 . 36 / 1 1 \ 

V 6-p - 5-p + ^'^^'-'^ • -no- W ■" 503) 

= 8,025 4/ ^'^^^'^ j^ = 3054- cubic feet. 

o,u^ay _ Q QQQ1222 + 0,0007549 ^ 



The mean of the values is 

C = ?5fi±.i2!i = 330 cubic feet. 



972 GENERAL PRINCIPLES OF MECHANICS. [§478. 

§ 478. In order to obtain the formula for. the depth of the 
water, let us put the upper depth = a^ and the lower = a^, the slope 
of the bed = a, and consequently the fall of the bed — I sin. a. 
The fall of the stream is then 

h = a^ — ax + I sin. a ; 
hence we have the equation 

whence we deduce 



I 



_ (1 1\Q^ 

^ ""' W F:) 2 g 

^ F, + F, \F^' F,'f 2 g 



By the aid of this formula we can determine the distance /, 
which corresponds to a given change a^ — a^ in depth. If the in- 
verse problem is to be solved, we must resort to the method of 
approximation, i.e., we must calculate first the lengths li and l^ cor- 
responding to the assumed changes a^ — a^ and aj — a^ of depth, 
and then we must find by a proportion the change of depth corre- 
sponding to the given length I (see Ingenieur, Arithmetic, § 16, 
V, page 76). 

The formula can be simplified, when the width h of the stream 
is constant. In this case we can put 

/I 1 \ Q^ _ F:- F,^ Q^ ^ (F, - F,) {F, + F, ) V 

\F^ F; )2g F,'F; ' 2g F,' ' 2g 

= K^«lH^^±^ . ^ approximatively = 2 ^^^-^ . ^ 
ax Z g a^ A g 

and in like manner 

V /I 1 \ e^ _p{f:-\- Fx^) v: 



\ \f: ^ FxV 2 g 



F, + Fx \F; ^ FxV %g (i^o + ^i) Fx'' %g 

p V ^ 
approximatively = ^^t . ^, from which we obtain 

a^o Z g 



I 



and consequently 



(«, _ ^.) ^1 _ A . 1^) 

L, . -^- . -^ s%n. a 

a,h %g 



a^— ax _ a^o % g 

a,' 2g 



§ 479.] THE MOTION OF WATER IN CANALS AND RIVERS. 973 

By the aid of this formula we can obtain directly, for a given 
distance /, the corresponding change {a^ — a^} of depth of the stream. 

Example. — A horizontal ditch 800 feet long and 5 feet wide is required 
to convey 20 cubic feet of water per second ; the depth of water at the 
entrance is 2 feet, what will be its depth at the end of the ditch ? Let us 
divide the entire length of the ditch into two equal sections and determine 
by the last formula the fall for each of them. 

Here sin. a = 0, I = -— = 400 and 5 = 5; for the first section, v^ = 

20 

= 2 ; hence C = 0,00812 and ag = 2; now since p = 8J, it follows that 



S. 5 



^0,00812 ^'^ ^ 



«o — ^1 



10 ' 2g\ ,^^ 2,7608 ^,^, ^ 

^ ' . 400 = -\-^zr = 0,188 feet. 



2 4 I 15,1 



V 2*2^ 

Now for the second section a^ = 2 — 0,183 = 1,817 and p is about 

20 
= 8,2, v^ = ^ „Q^ = 2,201 ; the fall in the second section will be 

(^..^.^ 8,2 2,20in 
^— aj^o/- • *«" = o;9l72 = O'^"" ' 
^ 1,817' 2g I 
hence the entire fall is 

= 0,183 + 0,240 = 0,423 
and the depth of water at the lower end is 

= 2 — 0,423 = 1,577 feet = 18,92 inches. 

§ 479. Floods and Freshets. — When the water level in 
rivers or canals changes, it is accompanied by changes in the ve- 
locity and in the quantity of water carried. A rise of the water 
level not only increases the cross-section, but also causes a greater 
velocity and, therefore, for a double reason a greater discharge ; in 
like manner a fall of the water level causes both a diminution of 
velocity and of cross-section, and consequently a two-fold diminu- 
tion of the quantity of water. If the original depth = a and the 
present one = a^ and the upper width of the canal = h, we can 
put the increase of the cross-section = h (ai — a) ; hence the 
cross-section, when the water level has risen a distance «i — a, is 
Fi = F -{- b (ai — a) and consequently 

■^1 _ 1 , ^ {ai - a) 
F-^^ F ~' 
and we can put approximatively 



/ 



^ ^ -, b (a, - a) 
F ^ 2 F 



974 GENERAL PRINCIPLES OF MECHANICS. [§479. 

If the original wetted perimeter =: ^, the present one = ^i and 

the angle of slope of the banks = 6, we have 

2 (ai — a) , 
Pi = p -\- — ^.- — 7^, whence 

-^ -^ sm. u 

i^ = 1 + ^.HJ^ and 

p p sm. u 

a/ P\ -, a^ — a 
y -- = 1 + — ^-^, or 
p p sm. 

J p_ _ ^ _ a, — a 
Pi p sin. 6' 

Now the Telocity for the original depth of water is 

c = 92,26 y — J, and for the present depth it is Ci= 92,26 y — .j; 
hence 



z= i/^ \/jL - /l 4_ ^ (^1 - ^) \ /l _ ^1 - ^ \ 

^ F ' ^ p, \ '^ ^ F )\ p sin. 6/ 

= 1 ^ (ai — a) l^—f^ — — -. — ^l, 

^ ^ \%F p sm. dr 



1\ 

and the relative change of velocity is 

1) '-^'- = («, - a) (^^ - ^4^). 

On the contrary, the ratio of the quantity of water carried by 
the river is 

Q F c \ F /L ^ ' yl F p sm. 6/ J 

^ ^ \2 F p sm. 6/ 

and the relative increase in the quantity of water is 

^ Q ^ ^ \2 F p sm. 01 

We can put less accurately, but in many cases, particularly for 
wide canals with little slope, sufficiently so, F — a b and neg- 
lect — ^— 7c, in which case we have more simply 
p sm. 6 ^ -^ 

Cx — G _ ^ ai — a . Qi—Q_oCti — a 

— n and — pr — — 1} . 

c ^ a Q ^ a 

According to these formulas the relative change in the velocity 
is half as great and that in the quantity of water ^ as great as the 
relative change in the depth of the water. 

The foregoing formulas are only applicable, when the motion 



479.] THE MOTION OF WATER IN CANALS AND RIVERS. 



975 



of the water in its channel is permanent, in which case the depth 
of the water is constant, but they do not hold good when the depth of 
the water is variable. The mean velocity in the same transverse sec- 
tion is greater, when the water is rising, and smaller, when the 
water is falling than when the depth of the water is constant ; 
hence in the first case more water and in the second case less water 
passes through than when the motion is permanent. 

Example — 1) If the depth of the water increases ^l^ the velocity is in- 
creased ^ and the quantity of water -^ of its original value. 

2) If the depth decreases 8 per cent., the velocity is diminished 4 per 
cent, and the quantity of water 12 per cent. 

3) By the aid of the more accurate formula 

Q - '^^i ^^\2F psin.dj 
we can construct a water-level scale K M, Fig. 809, upon which we can 
read off the quantity of water passing in a canal for any depth K X, when 

we know the quantity of water for a certain mean 
depth. If 5 = 9 feet, d^ = S, a = d, and 6 = 45°, 
we have 

(9 -f 3) 3 



*)• 



Fig. 809. 




F = 



2 



— == 18 square feet, 



Qt- Q / 3 . 9 



Q 



=( 



2.18 



3 + 2 . 3 . V2 = 11,485, and 
0,707; hence 
) («! — a) = (0,750 — 0,123) (a^ — a) 



P 



sin. 6 = ^/^ 



11,485 . 0,707y 
= 0,627 («i — a). 

If the quantity of water corresponding to the mean water level is Q = 
40 cubic feet, we have 

Qj = 40 + 40 . 0,627 {a^ - a) = 40 + 25 {a^ - a). 

If «! — a = 0,04 feet = 5,76 lines, Q^ = 41 cu. ft. ; if a^ — a — 0,08 
feet = 11,52 lines, Q^ = 42 cu. ft. ; \ia^—a = — 0,04 feet, q^ = 39 cu. ft., 
etc., a scale, whose divisions are X J!f = L JV = 5,76 lines apart, would 
give the quantity of water to a cubic foot. The accuracy of course di- 
minishes as the difference of the depth of water from the mean depth in- 
creases. 

Remark. — The construction of mill-races, canals for bringing water, as 
well as the location of dams, weirs, etc., will be treated of at length in the 
second volume. 

Final Remark. — The author has discussed at length the sutjject of the 
motion of water in canals and rivers in the Allgemeine Encyklapadie, 
Vol. n. Article '' Bewegung des Wassers in Canalen und Fliissen," and has 
given there a list of the treatises upon this subject up to 1844. Rittinger's 
tabulated synopsis of the experiments upon the motion of water in canals 
is contained in the "Zeitschrift des osterreichischen Ingenieurvereins," 
year 1855. 



976 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 480. 



CHAPTER VIII. 



HYDROMETRY, OR THE THEORY OF MEASURING WATER 



Fig. 810. 



§ 480. Gauging. — The discharge of a running stream within 
a certain time is measured either by gauged vessels, by a dis- 
charging apparatus, or by hydrometers. The most simple method 
is that by means of gauged vessels, but this is only applica- 
ble to small quantities of water. The vessel is most frequently 
composed of boards, and is therefore parallelopipedical in form, and 
to increase its strength, it is generally bound with iron hoops. The 
manner of calculating the exact contents of this vessel is given in 
the Ingenieur, page 208. The water is brought to the vessel by a 
trough E F, Fig. 810, at the end of which is placed a double clack, 

by means of which the water can 
be made to flow into the vessel or 
alongside of it. In order to deter- 
mine accurately the depth of the 
body of water, we employ a scale 
K L. If, before we begin the 
measurement, we lower the pointer 
Z until it touches the surface of 
the water, which, perhaps, may 
only cover the bottom, and read 
off on the scale the depth of the 
water, we obtain the depth Z Z^ of the water to be measured by 
subtracting the former reading from that given by the scale, when 
the pointer Z, after the completion of the observation, is brought into 
contact with the top of the water. The clack must of course be so 
arranged before the experiment that water shall discharge alongside 
of the vessel. If we are satisfied that the influx into the trough 
has become constant, we observe the time upon a watch held in the 
hand and turn the clack around so that the water will discharge into 
the vessel ; when the vessel is full, or partially so, we observe again 
upon the watch the time and return the clack to its original posi- 
tion. From the mean cross-section F and the depth Z Zi = s of 
the body of water, we calculate the total discharge V = F s, which. 




§ 481.] 



HYDROMETRY, ETC. 



977 




when divided by the duration of the influx, which is the difference 

t of the two times observed upon the watch, gives the discharge per 

second 

Fs 

Remark. — If we wish to know at any time the discharge of a variable 
stream of water, we can employ the apparatus represented in Fig. 811, 

which is often met with in salt 
Fig. 811. works. Here there are two meas- 

uring vessels A and B^ which are 
alternately filled and emptied. The 
water, which is brought to them 
by the pipe F^ passes through a 
short tube G G^ which is rigidly 
connected with the lever D E which 
turns about C If one of the ves- 
sels, as, E.G., J., is filled, the water 
flows through a small trough B. 
and fills the little bucket if, which 
then draws the lever down and the 
pipe G G comes into such a position as to carry the water into B. The 
valves and P are opened by cords passing around pulleys and attached 
to the lever. The opening of the valves is assisted by iron balls, which 
give the last impulse to the lever when it is descending. The buckets M 
and iV contain small orifices, through which they empty themselves after 
the lever has turned. A counter attached to the apparatus gives the num- 
ber of strokes, which can be read off at any time. Other apparatuses of 
the same sort, which were employed by Brown, are described in Dingler's 
Polyt. Journal, Vol. 115. In reference to a new apparatus for measuring 
water by Noeggerath, see Polyt. Centralblatt, 1856, No. 5. Compare also 
the works of Francis, Lesbros, etc., which have been cited. See also further 
on, § 506. 

§ 481. Regulators of EiHux. — Very often small and medium 
discharges are measured by causing them to pass through a hnown 
orifice under a known head. From the area F of the orifice and 
the head h we determine, by the aid of the coefiScient of efflux, the 
discharge per second 

Q = 11 F V%gh. \ 

Poncelet's orifices are the best for this purpose ; for their coeffi- 
cients of efflux are known (§ 410) with great accuracy for different 
heads, but they are only applicable, when the discharge is a medium 
one. The author employs in his measurements of water four such 
orifices, one 5, one 10, one 15 and one 20 centimeters high and all 
62 



978 



GENERAL PRINCIPLES OF MECHANICS. 



[§48t 



Fig. 813. 
A 17 




20 centimeters wide. These orifices are cut out of brass plates, 
which are placed upon wooden frames such as A C, Fig. 812, and 
these frames can be fastened to any wall by means of four strong iron 
screws. But in many cases we must employ much 
larger orifices for which the coefficients of efflux have 
not been determined so certainly ; and very often we 
can only employ overfalls or notches, which arc even 
less accurate. But in any case we should cndcLivcr 
to produce both perfect and complete ccRtriicliou. 
Hence, if the orifice is in a thick wall, we should bevel it off upon 
the outside. The corrections to be applied for partial rud incom- 
plete contraction have been sufficiently explained in §§ 416, 417. 

In order to measure the quantity of water in a trough, we first 
put the mouth-piece in its place and then wait until the head 
becomes permanent. In order to measure the head, we can em- 
ploy either the fixed scale K L with a pointer. Fig. 813, or the 
movable one E F^ Fig. 814. If we wish to observe the efflux directly 

Fig. 814. 





Fig. 815. 



at the sluice gate, it is advisable to attach to the guides two brass 
scales B C and D E, Fig. 815, with the pointers F and G by means 
of which we are able to read ofi" with more cer- 
tainty the height of the orifice. It is always bet- 
ter, when measuring water, to employ a new 
sluice gate and new guides which are properly 
beveled outwards. 

The most simple way of measuring the water 
in a trough is to place a loard OF, Fig. 701, 
beveled at the top, across it and to measure the 
overfall which is produced. If the ditch or trough is long and 
nearly horizontal, considerable time will elapse before the flow be- 
comes permanent, and it is, therefore, advisable before beginning 




1 482.] 



HYDROMETRY, ETC. 



979 



Fig. 816. 




the measurement to put in another board, which will prevent for 
some time the efflux of the water and thus hasten its rise to the 
height necessary for a permanent flow. 

In order to measure the discharge of a creek, we can construct 

a dam A B, Fig. 816, of boards and 
allow the water to flow through an 
opening C in it, or we can employ a 
simple overfall or weir (this subject 
will be treated more at length m the 
second volume). 



Remark. — The most simple method 
of determining the head is to observe the 
position of the pointer, first, when its point touches the surface of the 
water, while the flow is permanent, and secondly, when it touches t^ie sur- 
face of the still water which is on a level with the top of the sill. The 
difference of the two observed heights is the head of water or the height 
of the water above the sill. We must be careful in observing the last 
height of the pointer to pay attention to the action of the capillary attrac- 
tion, in consequence of which the level of the water may be 1,37 lines 
above or below the sill, before efflux over the latter will begin (see § 380). 

§ 482. "We can easily measure the quantity of water carried by 
a oanal or trough A B, Figs. 817 and 818, in the following man- 

FiG. 818. 



Fig. 817. 




Affi 



^Y,yY.\\v.\V-SNNVv"Vv- ■ NN SN-XSN-V^-VN SvyV-NV vC~.VS-XVNSVy-C 




~^^mi 



.^^\^\\^^\^\\N^\^^>^'»^^»^\y^■;»i^t^»;«y 'JV^} ^^«t^^^ 



ner : a board, the lower edge of which has heen beveled^ is inserted 
in the trough in such a manner as to leave an opening between it 
and the bottom of the latter, through which the water will pass. 
This method has an advantage over that in which overfalls are 
employed, viz. : the water, which is dammed back, comes to rest 
better, and we can, therefore, measure the head more accurately. 
AYhen it is possible to have a free efflux, as in Fig. 817, we should 
prefer it, since greater accuracy can be obtained, but when the 



980 GENERAL PRINCIPLES OF MECHANICS. [§ 483. 

quantities of water are large, it is not possible to prevent the water 
from rising, and we are obliged to be satisfied with an efflux 
under water, such as is represented in Fig. 818. If the trough 
ends but a short distance from the orifice, i.e., if it forms a sJioot, 
the water flows through it almost freely and we have one of the cases 
of Lesbros' experiments (§ 418). If a denote the height and h the 
width of the orifice, h the head measured to the middle of the ori- 
fice and (i the coefficient of efflux, taken from Table II, § 418, we 
have the discharge 

Q = PL ah V% g h. 

If, on the contrary, the trough is long, or if the water, which is 
flowing away, is so obstructed that its surface becomes horizontal, 
the water will pass all portions of the cross-section of the orifice 
with the same velocity, which is that corresponding to a head equal 
to the difference of level of the water A above and the water B 
below the orifice, and we have only to substitute in the latter 
formula for li the difference of level. 

If the water flows into the air, or if the surface of the lower 
water, as in Fig. 817, does not rise above the upper edge of the 
orifice, we must substitute for an orifice with beveled or with 
rounded edges 

[i = 0,965, 
and consequently, when the depth of the stream is a and its width h, 

Q = 0,965 a l VWgJi, 
or more accurately, when a^ is the depth of the approaching water 
and a that of the water flowing away (see § 398), 



Q = 0,965 «& 



^^_9± 



1 



\aj 



When the efflux takes place under ivater, in which case the lower 
surface of the water is above the upper edge of the orifice (see Fig. 
818), an eddy is formed behind the wall of the orifice, by which the 
efflux is considerably interfered with. According to several experi- 
ments of the author, for an orifice with a sharp edge we must put, 
as a mean value, ^i = 0,462, 

and, on the contrary, when the edge is rounded off in the shape of a 
quadrant, fi = 0,717. 

Example. — ^In order to find the discharge of a trough A B, Fig. 818, a 
sharp-edged board C D was placed in it and an efl3^ux under water was 
thus iDroduced ; the following observations were then made. Width of 
orifice or trough & = 3 feet, height of orifice or distance (7 £^ of the edge G 



§483.J 



HYDROMETRY, ETC. 



981 



of the board above tbe bottom of the trough a = Q inches, length of the 
pointer Z above the orifice h^ = 0,445 feet, length of the pointer Z^ below 
the orifice Ag = 1,073. Hence the difference of level is 

h = ?i^ - h^ = 1,073 - 0,445 = 0,628 feet, 
and the required discharge is 

Q = 0,462 . 8,025 . 3 . 0,5 VAg -T^ = 5,56 Vo;628 = 4,40 cubic feet. 

§ 483. If the coefficient of efflux were always the same for sim- 
ilar cross-sections, the triangular or two-sided notch ABC, Fig. 819, 
would have a great advantage over the notch with a horizontal sill ; 

but this assumption, as we have seen in the 
case of circular apertures, is not correct for 
small orifices, and only approximative^ so for 
large ones. Professor Thomson, of Belfast, 
recommends such notches as useful for measur- 
ing water. From the width A B = b and the 
height C D = li, w^e obtain the discharge 

Q = T% ^' ^371 (see § 402), 
and if we put, with Prof. Thomson, the coefficient of efflux fi = 0,619, 



Fia. 819. 







Fig. 820. 



Q = 0,33 -^ V2gh = 0,132 b h' cubic feet. 

Orifices, so shaped that the discharge through them shall be 
proportional to their height, are useful in measuring water. If they 
are provided with a sluice-gate the height of the opening is the 
measure of the discharge. Let the head above the upper edge of 
such an orifice A B C D, Fig. 820, be ^ — A, the length of this 

edge \)Q A B = h, fchat of the lower edge, 
C D = h^, and the height of the orifice, 
A D ^^ a. Horizontal lines at the distance 

- from each other will divide this orifice 
n 

into strips of equal height, and the dis- 
charge — through each of them should be 

the same. For the upper strip, whose width 
is b and for which the head is h, we have 

n n 




V2 g h, 



and, on the contrary, for another strip at a distance M = x be- 
low the surface of the water, whose width M N = y, 



982 GENERAL PRINCIPLES OF MECHANICS. [§484 



n n 

equatinsr these two values of — , we obtain 

. n 

yVx = b Vh, or - 

' X 

The curve B JV C, which bounds the orifice on the side, belongs 
to one of the system of curves discussed in Article 9 of the Intro- 
duction to the Calculus; its asymptotes rire the horizontal line 
I'and the vertical one X. 

From Q, li and a we obtain 

Q 



1) the upper width of the orifice h 



^V2gh' 



/'. 



2) the width of orifice at the depth x,y — t 

3) the lower width of the orifice hi = l \j 
The area of the orifice is 



+ a 



F^ ^h{Vh {h + a) - h), 
and consequently the mean head is 

~^g\Ff \ s/% (7^ + a) _ iJ ' 2* 
If the orifice is provided with a sliding gate, when it is raised a 
distance I) M = ai, it gives an orifice of efi9.ux M (7, the discharge 

through which is Qi = — Q. 

§ 484. Prony's Method. — As considerable time often elapses 
before the flow of the water, which has been dammed back, be- 
comes permanent, the following method, 2Jro2:>osed hy Prony, can 
often be employed with advantage. We begin by closing the 
orifice completely by means of a sluice-gate, and we wait until the 
water has risen to a certain height, or as high as circumstances 
will permit ; we then open the gate enough to allow more water to 
be discharged than is arriving, and we measure the height of the 
water at equal intervals of time, which should be as small as pos- 
sible ; finally, we close the orifice again perfectly and observe the 
time ti in which the water rises to its former height. Now during 
the lapse of the time t + tx the same quantity of water has of 
course arrived and been discharged ; hence the quantity of water 
which arrives in the time ^ + ^1 is equal to the discharge in the 



§485.] HYDROMETRY, ETC- 9^3 

time t. If the heads, while the level of the water was sinking, were 
Aj,, hif ^2, ^3, and hi, we have the mean velocity 

V = ^^ ( Vy, + 4 V i; + 2 /a; + 4 VT3 + VT,) (see § 453), 

and if the area of the opening of the sluice is F, the discharge in 
the time Hs 

V = fill^ll ( /X^ + 4 /i; + 2 fT, + 4 4/1; + V¥,); 
hence the quantity/ of luater arriving in a second is 

Example. — In order to measure the quantity of water in a brook, which 
we wish to employ to turn a water-wheel, the stream was dammed up by a 
wall of boards, as is represented in Fig. 816, and after opening the rec- 
tangular orifice in it, we made the following observations : initial head, 2 
feet; after 30", 1,8 feet; after 60", 1,55 feet; after 90'', 1,3 feet; after 120", 
1,15 feet; after 150,", 1,05 feet; and after 180", 0,9 feet; width of the ori- 
fice = 2 feet, height = ^ foot, time required for the water to rise to former 
level 110". In the first place the mean velocity is 

8 025 - 

V = -V^ (V2 + 4 V'1,8 + 2 Vl,55 + 4 V 1,3 + 2 V1X5 + 4 Vl,05 + V0,9) 

= 0,4458 (1,414 + 5,364 + 2,490 + 4,561 -f 2,145 + 4,099 + 0,949) 

= 0,4458 . 21,022 = 9,372 feet. 

But F = % . -|- = 1 square foot, the theoretical discharge is, therefore, 

= 9,372 cubic feet. If we assume that the coefficient of efiiux = 0,61, we 

obtain the required quantity of water 

0,61 . 180 
Q = T^ TT-A • ^,^'^^ = 3,548 cubic feet. 

§ 485. Water-inch.- — When we are required to measure small 
quantities of luater, we often allow it to discharge under a given 
head through circular orifices in a thin plate, which are one inch 
in diameter. We call the discharge through such an orifice, under 
the smallest pressure, i.e. when the surface of the water is one line 
above the uppermost part of the orifice, a water -inch (Fr. pouce 
d'eau ; Ger. Wasserzoll or Brunnenzoll). The French assume that 
a water-inch (old Paris measure) corresponds to a discharge in 24 
hours of 19,1953 cubic meters, or 

in 1 hour, 0,7998 cubic meters, and 
in 1 minute, 0,01333 cubic meters ; 
but the older data, given by Mariotte, Couplet, and Bossut, differ 
considerably from the above. According to Hagen, the water-inch 
Cfor Prussian measure) discharges 520 cubic feet in 24 liour,«, cv 
0.3G11 cubic fe?t in a minnto. Pronv's double ivater modulr'^ ■:• 



984 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 485. 



Fig. 821. 



" nouveau pouce d'eau"), which corresponds to an orifice 2 centi- 
meters in diameter, under a pressure of 5 centimeters, and which 
discharges 20 cubic meters in 24 hours, has not been adopted gen- 
erally. 

The observations can be made with more certainty when we 
have a greater head ; it is simpler to make this head equal to the 
diameter 1 inch of the orifice. According to Bornemann and Ro- 
ting, such a water-i7ich passes daily 642,8 cubic feet (Prussian) of 
water (see the Ingenieur, page 463). 

The apparatus, by which we measure the water with the aid of 
the water-inch, is represented in Fig. 821. The water to be meas- 
ured is discharged from the 
pipe A into a box B, from 
which it passes through 
holes, made in the parti- 
tion G D below the level 
of the water, into the box 
E ; from it the water is 
discharged through circu- 
lar orifices F one inch in 
diameter, which are cut 
out of sheet iron, into the 
reservoir G. To preserve the level of the water 1 line above the 
top of the orifice we must have a sufficient number of holes, a por- 
tion of which are closed by stoppers. We employ for more accu- 
rate determinations in addition the orifice F^ which allows J, \ of 
a water-inch to pass through. When the quantity of water is very 
great, we divide it into several portions and measure in this way 
but one portion, as, e.g., a tenth. This division is easily accom- 
plished by conducting the water into a reservoir with a certain 
number of orifices on the same level and catching the water deliv- 
ered from one of the orifices only in the above apparatus. 

Remark. — We can also employ cocks and other regulating apj)aratuses 
for measuring water, when we know the coefficients of resistance corre- 
sponding to every position. If h is the head, F the cross-section of the pipe 
and // the coefficient of efflux for the cock, when fully open, we have the 
discharge 




or inversely 



H = 



Q 

^V2" 



^^^^7 = (f)-'^'- 



486.] 



HYDROMETRY, ETC. 



985 



Now if we put the coefficient of resistance for a certain position of the 
cock, which may be taken from one of the tables given previously, = C, 
we have the corresponding discharge 

2>gh fiF^/^gh Q 



Qx 



F 



\ 



1 

Q 



Vl + ^^ C Vl + ;.^ c 



"We can construct from the above formula a convenient table, and we 
have only to glance at it when we wish to know the discharge correspond- 
ing to a certain position of the cock. If, e.g., fx = 0,7 and F^ = A: square 
inches, we have 



«i = 



0,7 . 4 . 12 . 8,025 VA 



Vl + 0,49 C 

or, if h is constant and = 1 foot, 

_ 269,64 
V — 



= 269,64 y -^^^^^ ^ cubic inches, 



Vl + 0,49 C 

Now if the cock is turned 5°, 10^ 15°, 20°, 25°, etc., the coefficients of 
resistance are 0,057, 0,293, 0,758, 1,559, 3,095, and the corresponding dis- 
charges are 266, 252,1, 230,2, 203, 170 cubic inches. 

§ 486. In order to regulate the efflux through an orifice F, 
Fig. 822, we employ either a cock or valve A, Fig. 822, which is 
Fm. 822. Fig. 823. 





regulated by means of a lever and a float K, so that the same quan- 
tity of water is discharged through B as through F. 

The discharge of water from a reservoir B D E, Fig. 823, 
through a lower orifice or tube D can be regulated by means of a 
wide overfall B, since a moderate change in the quantity of water, 
discharged through A, will produce but a slight change in the 
height of the water above the sill B ; hence the augmentation of 
the head of the orifice of efflux will be inconsiderable. 

Let i^ denote the area of the orifice D, h the height of the sill 
of the overfall above the middle of the orifice and h^ the height of 



986 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 487. 



the surface of the water above the same sill. We have the dis- 
charge through D 

Q = II FV%g{li +X), 

when the coefficient of efflux is [i. Substituting the head hx above 
the weir, which can be determined from the discharge Q^, the width 
hi and the coefficient of efflux //i by means of the equation 



Qx = % fii hi Vi g hi, or by the formula 

we obtain the expression 

from which it is easy to see that Q varies less with Q^, the greater 
the value of h is and the greater the width h^ of the overfall is. 

The Avidth h of the overfall can be easily increased by giving it 
a curved form like BOB, Fig. 824. The discharge through the 

orifice D is then almost 
Fig. 824. constant, although the 

quantity of water flow- 
ing in may be very va- 
riable ; for the height of 
the water above the long 
curved sill is always 
small compared with the 
height of this sill above 
the orifice of efflux. 

Remakk.— Such an apparatus for dividing the water was constructed 
of sheet iron for the Wernergrdben at Freiberg by Oberlcunstmeister Schwam- 
krug. It discharges through a rectangular orifice I>, which is 5 feet long 
and 1 foot high, almost invariably 40 cubic feet of water per second, while 
the remaining water passes over the overfall, the sill of which lies 2 feet 
above the uj^per edge of the orifice, and flows on in the ditch to where it 
is wanted. 

§ 487. Hydrometric Goblet. — We can employ to measure 
small quantities of running water the small vessel, represented in 
Fig. 825, which I have called the hydrometric gohlet. This instru- 
ment consists of a tube B, 12 inches long and 3 inches in diameter 
with a funnel-shaped mouth -piece A, and of a vessel D, 6 inches 
high and 6 inches wide, which is united to B by an intermediate 




§ 487.] 



HYDROMETRY, ETC. 



987 



Fig. 825. 



conical piece C. This vessel has an orifice L L in the side, in 
which we can insert mouth-pieces containing different sized circu- 
lar orifices in a thin plate. The instrument is held by means of 
the handle H under the water S, which is being discharged, E.G., 

from the pipe R, and the water thus caught 
is allowed to discharge itself through the ori- 
fic'es L L. In order to tranquilize the water 
in the vessel a fine sieve or wire gauze is 
placed in the reservoir D, and in order ' o ob- 
serve the head of the water a glass tube P, 
which is placed close to a brass scale and ends 
a half an inch from the bottom of the vessel, 
is added to it. From the observed head, the 
known cross-section of the orifice and the 
corresponding coefficient of efflux, we can cal- 
culate the discharge by means of the formula 

Q = fi F V% g h. 

If we prepare a small table, we can spare 
ourselves this calculation and the only opera- 
tion, which we are required to perform, is a 
simple interpolation between the values in the 
table. If d is the diameter of the orifice, 




F = 



IX d' 



, and therefore 



[l IT 



Q=^^d' V2gh = ^-^ V¥~g. d' Vh. 






_ The discharge Q is double, when the cross-section or d"^ is double, 
or when the head is four times as great. If we so arrange the in- 
strument that the maximum head shall be four times the minimum ; 
if, E.G., the former is 12 and the latter 3 inches, and if we employ a 
series of orifices whose diameters form the geometrical series 

d, V'i d,^ d,^ V2 d, 4 d, etc. 
I.E. d, 1,414 d, 2 d, 2,828 d, 4 d, etc., 

we obtain a means of measuring all discharges from the minimum 
given by the smallest orifice with the diameters? under the smallest 
head, to the maximum, given by the largest orifice with the diam- 
eter VrT. d under the greatest head 4 h. 



988 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 487. 



If we assume for 





L 


II. 


in. 


IV. 


V. 


VL 


VII. 


d = 


1 
= 0,^350 


= 0,1768 


1- 
= 0,2500 


= 0,3535 


1 

= ojooo 


1 V2 
= 0,7071 


1 inch 
= 1,0000 


fX = 


0,690 


0,675 


0,660 


0,647 


0,635 


0,627 


0,620 



we can calculate the following useful table. 

Table of the hourly discharge in cubic feet for the following orifices. 



Head h in inches. 


I. 


II. 


III. 


IV. 


V. 


VI. 


VIL 


3 


0,85 


1,66 


8,25 


6,87 


12,51 


24,70 


48,85 


4 


0,98 


1,92 


8,75 


7,86 


14,44- 


28,52 


56,40 


5 


1,10 


2,14 


4,19 


8,23 


16,15 


31,89 


63,06 


6 


1,20 


2,85 


4,60 


9,01 


17,69 


34,93 


69,08 


7 


1,80 


2,54 


4,96 


9,78 


19,10 


37,78 


74,61 


8 


1,39 


2,72 


5,81 


10,41 


20,42 


40,33 


79,77 


9 


1,47 


2,88 


5,63 


11,04 


21,66 


42,78 


84,60 


10 


1,55 


8,08 


5,98 


11,65 


22,84 


45,09 


89,18 


11 


1,68 


3,18 


6,22 


12,20 


23,95 


47,30 


93,58 


13 


1,70 


3,32 


6,50 


12,74 


25,01 


49,40 


97,69 


18 


1,77 


8,46 


6,77 


18,26 


26,04 


51,42 


101,68 



The manner of using this table is shown by the following 
example. 

Example. — In order to determine the quantity of water furnished by a 
spring, the water from it was caught in a hydrometric goblet, and it was 
found that a state of permanency occurred when the efflux took place 
through the orifice V (one half inch in diameter) under a head of 10,4 
inches. According to the table for A = 10 inches 

Q = 22,84 cubic feet per hour, 
and for A = 11 inches 

Q — 23,95 cubic feet, 

consequently the difference for one inch is 1,11 cubic feet, and for 0,4 inches 



§488.J HYDROMETRY, ETC. 989 

0,4 . 1,11 = 0,444, Hence the discharge under the head h = 10,4 inches is 
V^ q = 22,84 + 0,444 = 23,284 cubic feet. 

§ 488. Floating Bodies. — The discharge of large creeks, 
canals and rivers can only be measured by means of hydrometers, 
which indicate the velocity. The simplest of these instruments are 
floating bodies (Fr. flotteurs ; Ger., Schwimmer). We can use any 
floating body for this purpose, but it is safer to employ bodies of 
medium size and of but little less specific gravity than the water 
itself. Bodies whose volumes are about j'o of a foot are quite large 
enough. Very large bodies do not easily assume the velocity of the 
water, and very small bodies, particularly when they project much 
above the level of the water, are easily disturbed in their motion by 
accidental circumstances, such as the wind, etc. A simple piece of 
wood is often employed, but it is better to cover the wood with a 
light-colored paint; hollow floats, such as glass bottles, sheet- 
iron balls, etc., are better; for we can fill them partially with water. 
Floating balls are, however, most generally employed. They are 
made of sheet brass and are from 4 to 12 inches in diameter; to 
prevent their being lost sight of, they are covered with a coat of 
light-colored oil paint. Such a floating ball A, Fig. 826, gives the 
velocity at the surface only, and often only that in the axis of the 
stream. By uniting two balls A and B, we can find also the 
velocity at different depths. In this case one ball, which is to be 
submerged, is filled with water, and the other contains enough to 
prevent more than a small portion of it from projecting above the 

level of the water. 

Fig. 826. Fig. 827. r^j^^ ^wo balls are 

united by a string, 
wire or thin wire 
chain. We first de- 
termine by a single 
ball the superficial 
velocity c^, and we 
then determine the 
mean velocity c of the two connected balls ; now if we denote the 
velocity at the depth of the second ball by Ci, we can put 

c = -^ — -, and, therefore, inversely, ^ = 2 c — c^. 

If we unite the balls successively by longer and longer pieces 
of wire, we obtain in this way the velocities at greater and greater 





990 GENERAL PRINCIPLES OF MECHANICS. [§489. 

depths. The mean velocity of a perpendicular is determined by 
allowing the second ball to swim near the bottom and putting 

_ c, + Ci 

it is more accurate, however, to take the mean of all the observed 
velocities in the perpendicular as the mean velocity. 

To obtain the mean velocity m a perpendicular, a floating staff 
Ai B^, represented in Fig. 828, is often employed, and it is very 
^ convenient, when it is used for meas- 

urements in canals and ditches, to have 

^sm^. ^ ^ ,--^ ^ g ---..^^— it made of short pieces which can be 

^^^^K^^S?^4^^^^ screwed together. The one used by 
^^i^i^^^^Ei^^^;^;^^^ the author is composed of 15 hollow 
^-|^^^-^^-— -.-.^^^ll ^^^^^l ^ pieces, each one decimeter long. In 
^^^^^IL^y^^ g g^^ g l^ order to make it float nearly perpen- 
^^U^^ SS -t^^ K^ dicularly, the lower part is filled with 
^ ^^^— =^^^- : :^-^^=^-^^^=^ enough shot to prevent more than the 

head from projecting above the water. 
The number of pieces to be screwed together depends, of course, 
upon the depth of the canal. 

We observe, when using the floating staff and the connected 
balls, that, when the movement of water in channels is not impeded, 
the velocity at the surface is greater than that at the bottom ; for 
the top of the staff and the uppermost ball are always in advance. 
It is only when the channel is contracted, as, E.G., by piers of 
bridges, that the opposite phenomenon is observed. 

Remark. — Generally, and particularly with large floating bodies such 
as ships, etc , the velocity of the floating body is somewhat greater than 
that of the water ; this is owing less to the fact that the body, in floating, 
slides down an inclined plane formed by the surface of the water, than to 
the fact that it does not participate, or at least only partially so, in the ir- 
regular internal motion of the water , this variation is, however, so slight, 
when the floatinnr bodies are small, as to be nesrliffible. 

§ 489. Determination of the Velocity and of the Cross- 
section. — We find the velocity of a floating ball by observing by 
means of a good watch with a second-hand or by means of a half- 
second pendulum (§ 327) the time t, in which it describes the dis- 
tance A B — s, Fig. 829, which has been previously measured and 
staked off on the shore. The required velocity of the sphere is then 

c — ,. In order that the time t shall correspond exactly to the 
t 



^ 489.J 



HYDROMETRY, ETC. 



991 



Fig. 829. 




distance measured on the shore, it is necessary to put two rods C 
and D^ by means of a suitable instrument, in such a position upon 

the other side of the river that the 
lines C A and D B shall be perpen- 
dicular to A B. Placing ourselves 
behind A, we note the instant the 
float K, which has been placed in the 
water some distance above, arrives at 
the line A C, and then passing be- 
hind B, we observe upon the watch the instant that the float ar- 
rives at the line B D\\i^ subtracting the time of the first observa- 
tion from that of the second, we obtain the time t, in which the 
space s is described. In order to determine the discharge Q = F c, 
we must know, besides the mean velocity c, the area F of the cross- 
section. To find this area, it is necessary to know the width and 
the mean depth of the water. The depth is measured by a gradu- 
ated soundmg-rod A B, Fig. 830, the cross-section of which is 
elongated and the foot of which is formed by board ; when the 
depth is great, we can make use of a sounding -chain, to the end of 
which an iron plate is attached, which, when the measurement is 
being made, lies upon the bottom. The width and the abscissas or 
distances from the shore corresponding to the depths measured are 

Fig. 880. Fig. 831. 





easily found for canals and small creeks E F G, 
Fig. 831, by stretching a measuring chain A B or 
laying a rod, etc., across the stream. When the 
river is wide, we make use of a plane-table if, 
placed at a proper distance A from the cross- 
section F F, Fig. 8S2, to be measured. If a o 
upon the plane-table is the reduced distance A 
of the fixed points A and from each other, and if we have placed 
a om the direction A 0, and thus made the direction a f of the 
width, which had been drawn, previously to putting the plane-table 
in position, parallel to the line ^ F to be measured off, each line 
of sight towards the points E, F, G, etc., in the transverse profile 
cuts off upon the table the corresponding points e, /, g, and 



992 



GENERAL PRINCIPLES OF MECHANICS. 



[^ 490 



Fig. 832. 



a e, af, a g, etc., are the distances A E, A F, A G, etc., upon the 
reduced scale. When using the sounding-rod to measure the 
depth, it is, therefore, not necessary to measure the distance of 

the corresponding points from the 
shore ; for the engineer, who is at 
the plane-table, can sight at the 
sounding-rod, when it is placed in 
the line E F. 

Now if the width E F, Fig. 831, 
of a transverse profile is made up 
of the portions Z*!, ^2? ^3? etc., and 
if the mean depths of these por- 
tions are «i, a^, a^, and the mean velocities Ci, Cs, Cs, etc., we have 
the area of the cross-section 

F ^= ttihi 4- «2 ^2 + ^3 ^3 + • • •> 

the discharge 

Q ■= a^h^Cx + ^2 ^2 ^2 -\- a^h^Cz ■{- ,. ., 

and finally the mean velocity 

_ Q _ cti ii Ci + ac, ic^ Cc> + . . . 

F a^hx + a^hi + . . . 

Example. — Upon a pretty straight and constant portion of a river th« 
following observations were made : 






Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


At the centre of the divisions of the width 

the depths were 

the mean velocities were 


5 
3 

1,9 


12 
6 
2,3 


20 
11 

2,8 


15 

8 

2,4 


7 
4 
3,1 



The area of the cross-section is 
i^= 5 . 3 + 12 . 6 + 20 . 11 + 15 . 8 + 7 . 4 = 455 square feet, 
the discharge is 
^ = 15 . 1,9 + 72 . 2,3 + 220 . 2,8 + 120 . 2,4 -|- 28 . 2,1 = 1156,9 cubic feet, 

and the mean velocity is 

1156,9 



c = 



-r^ = 2,54 feet. 
455 ' 



§ 490. Woltmann's Mill or Tachometer.^The best hy- 
drometer is Woltmann's tachometer or Woltmann's Mill (Fr. Moulinet 
de Woltmann ; Ger. hydrometrisches Fliigelrad von Woltmann), 
Fig. 833. It consists of a horizontal shaft A B with from 2 to 5 



§ 490.] 



HYDROMETRY, ETC. 



993 



surfaces or vanes F, inclined to the direction of the axis ; when 
immersed in water and held opposite to the direction of motion, it 



Fig. 833. 




indicates by the number of its revolutions the velocity of the run- 
ning water. To enable us to count the number of revolutions the 
shaft has cut upon it a certain number of threads of an endless 
screw Cy which work into the teeth of a cog-wheel D, which indi- 
cates, by means of a pointer and figures engraved upon the wheel, 
the number of revolutions of the wheel F. As we often wish to 
register a great number of revolutions the shaft of the cog-wheel 
carries a pinion, which takes into another cog-wheel ^, upon which 
we can read off, as upon the hour-hand of a ^atch, multiples 
(e.g., five or tenfold) of the number of revolutions of the vanes. 
If, for example, both cog-wheels have 50 teeth and the pinion has 
10, the second wheel will turn one tooth, while the first moves five, 
or the shaft of the vane wheel makes five turns. When the pointer 
of the first wheel is at 27 = 25 + 2 and that of the second at 32, 
the corresponding number of revolutions of the vane- wheel is ' . 

= 32.5 + 2 = 162. , 
The entire instrument with a sheet iron vane is screwed to a 
63 



V 



994 GENERAL PRINCIPLES OF MECHANICS. [§490. 

pole, so that it may easily be immersed and held in the water. In 
order to prevent the gearing from turning except during the time 
of the observation, its shafts run in bearings placed upon a lever 
G 0, which is pressed down by means of a spring, so that the teeth 
of the first cog-wheel do not take into the endless screw except 
when the string G B is, drawn upwards. The number of revolu- 
tions in a given time is not exactly proportional to the velocity of 
the water ; hence we cannot put v =^ a . u,m which u is the num- 
ber of revolutions, «; the velocity and a an empirical number, but 
we must put 

V = v^ -{- a u, 
or more accurately 

V — v^ + a u -{- (i u^ , , ,, 
or still more accurately 

V = a u -\- Vv^^ 4- i3 w'j . • 

in which v^ denotes the velocity of the water, when it ceases to 
move the vanes, and a and j3 are numbers to be determined by 
expei^iment. The constants i\, a, (3 must be determined for each 
particular instrument. By their aid a single observation gives the 
velocity, but it is always safer to make at least two and then take 
their mean value as the true one. 

Example. — If for a tachometer v^ = 0,110 feet, a = 0,480 and /? = 0, 

then « = 0,11 + 0,48 u, and if we have found the number of revolutions 

of the fan to be 210 in 80 seconds, the corresponding velocity of the 

water is 

210 
V = 0,11 + 0,48 . -— = 0,11 + 1,26 = 1,37 feet. 

Remakk 1. — The constants v^, a and i3 depend principally upon the 
angle of impact, i.e., upon the angle formed by the surface of the vanes 
with direction of the motion of the water and also with the direction of 
the axis of the wheel. If we wish to make, when, the velocities are small, 
pretty accurate observations, it is advisable to make the angle of impact 
large, i.e., about 70°. It is also desirable to have vane-wheels of different 
sizes and of different angles of impact, so that when the depth or velocity 
of the water is greater or smaller we can employ one or the other. 

Remark 2. — If the tachometer had no resistance to overcome in turn- 
ing, the vanes A B, Fig. 834, would describe the space C C^ = G D 
tang. G D G^ while the water describes G D ; hence, if we denote by v 
the velocity of the water and by 6 the angle of impact G B = G D G^^ 



§491.] HYDROMETRY, ETC. 995 

we have under this supposition the mean velocity of rotation of the vane- 
wheel 
Fig. 834. Fig. 835. <d^ ='d tang. 6, 

A jl^L from which it is easy to see that, 

y^\ ^^P^^C^ when r denotes the mean radius of 

Q Cx /' •^ y- "^^^^^^ the vane- wheel, the number of revo- 
/\ / ^ ^^^^Jp^ lutions is 

/ ,''C rg^jf t)^ 1) tang. 6 

. •^ ~ 2 7T r ~ 2 7T r ^ 

g I and that, consequently, it is directly 

H proportional to the velocity v of the 

water and to the tangent of the angle 
of impact and inversely to the mean radius of the vane- wheel. 

Remark 3. — In order to determine the superficial velocity of water we 
also employ a small wheel made of metal, like the one represented in Fig. 
835, and we allow only the lower part to be immersed in the water. The 
number of revolutions is given by a train of wheels, exactly as in the 
tachometer. 

§ 491. In order to determine the constants or the coefficients 
0f a tachometer, it is necessary to liold the instrument in running 
water, the velocity of which is known, and to observe the number 
of revolutions. Although only as many observations as there are 
constants are required, yet it is safer to make as many observa- 
tions as possible, particularly with very different velocities, and 
to employ the method of the least squares (see Introduction to the 
Calculus, Art. 36) and thus do away with the accidental errors 
of observation. The velocity of the water may be determined by a 
floating sphere, or we may catch the w^ater in a gauged vessel and 
divide the quantity of water caught by the cross-section. If the 
floating sphere is employed, the air must be still and the water must 
move uniformly and in a straight line. The vane-wheel must be 
immersed at several points along the path described by the floating 
sphere, and to insure perfect accuracy, the diameter of the sphere 
should be about equal to that of the vane-wheel. 

The second method of determination by catching the water, in 
which the mill is immersed, in a gauged vessel possesses many 
advantages. For this purpose, and for adjusting hydrometers 
generally, it is very desirable to have at one's disposition a 
hydraulic observatory, which consists of a gauged vessel, a trough, 
and a discharging vessel or reservoir. We can then give the 
water any desired velocity; for we can regulate not only the 
entrance of the water into the trough, but also, by inserting boards, 



996 GENERAL PRINCIPLES OF MECHANICS. [§ 491. 

we can regulate at will the velocity in it. In making the observa- 
tion, we have but to insert the tachometer at different parts of the 
cross-section of the trough, to measure the depth of this section by 
a scale, and then to gauge the quantity of water, which has passed 
through in a given time (§ 480). The area of the cross-section is 
obtained by multiplying the mean depth by the mean width, and 
the discharge Q is calculated from the mean cross-section of, the 
receiving reservoir and the depth of the water, which has flowed 
into it, by means of the formula 

finally, from Q and F we deduce the mean velocity of the water 

"^ - F ~ Ft' 
The corresponding number u of revolutions of the vane-wheel 
is the mean of all the revolutions observed when we inserted the 
Instrument in different parts of the transverse profile. 

If by experiment we have determined a series v^y ^2? '^z^ etc., of 
mean velocities and the corresponding numbers of revolutions, we 
obtain, by substituting them in the formula 

V = v^ ■\- a u, 
or in the more accurate one 



as many equations of conditions for the constants v^, a, (3, as w« 
made observations, and we can find from them the constants them- 
selves either by employing the method given in Art. 36 of the In- 
troduction to the Calculus, or by dividing these equations into as 
many groups as there are unknown constants, and combining them 
by addition into as many equations of condition as are necessary 
for the determination of v^, a and /?. - 

If we assume the passive resistances of the instrument to be 
small enough to be neglected, we can put v = a ti and determine 
a by moving the instrument forward in still water and observing 
the number n = u t of revolutions made in describing the space 
s = V t; then 

_ V _ f^t __ s 

u ut n . 

Rbmabk — 1) If we employ the simple formula with two constants, we 
can put, according to the method of least squares, 

_ 2(y'')2(a;)-2( a; y)2(y) ^ 2 (a;'^) 2 (y) - S (a; y) 2 (a;) • 

*'» - 2 {x") 2 (y') - [2 {X y)Y 2 {x') 2 (y«) - [2 {x y)^ 



§491.] HYDROMETRY, ETC. 997 

1 u 

in which x = - and y = -. and the siiJ^n 2 denotes the sum of all the 

values of the same kind as that which follows it, e.g. 

,111 

2 ^ir) = — + — + —+.. ., 

^'i ^2 ^3 

1 w. 1 ^2 1 u^ 
I.{xy) = — .-- + — .— + — . — + ... 

^1 ^1 ^2 «'2 ^3 ^3 

Example. — We have observed with a small tachometer that for the ve- 
locities 

0,163, 0,205, 0,298, 0,366, 0,610 meters 
the number of revolutions per second were 

0,600, 0,835, 1,467, 1,805, 3,142, 
and we wish to determine the constants corresponding to this instrument. 
By the aid of the formula given in the Remark, we obtain, since 

^ (^) = oi + o;k + • • ■ = i^'^""' 

, , 0,600 0,835 
^^^) = 0-;i63 + P05 + ---=-^^'^^^' 

^ ^''^ = (07^63)^+ (ojos)^^ '-= «^'^^«' 

2 (2/^) = 105,233, and 

, , 0,600 0,835 

^ (^^> = ^M63? + TPOS? + ••• = ^^'^^1' 
_ 105,233 . 18,740 — 80 ,961 . 22,759 _ 129,5 _ 
*^« ~ 82,846 . 105,233 - (80,96iy " "2162 ~ ^'^^^ ^^^ 

" = ^' = «'™^= 

hence the formula for this instrument is 

V =z 0,060 + 0,1703 u. 
Substituting u = 0,6, we obtain 

V = 0,060 + 0,102 = 0,162 
u = 0,835 gives 

u = 1,467, 

u = 1,805, 

and finally, u = 3,142, 

V = 0,060 + 0,535 = 0,595. 

The calculated values therefore agree very well with the observed ones. 

Remark — 2) We can also, according to Lapointe, insert the tachometer 
in a cylindrical pipe, and thus obtain the velocity of the water flowing 
through it. The counting apparatus can be placed outside of the pipe 
and connected with the vane-wheel Vjy means of a shaft. Lapointe calls 
this instrument une tube jaugeur (see " Comptes rendues," T. XXV, 1848; 



V = 0,060 + 0,142 = 0,202 

V = 0,060 + 0,249 = 0,309 

V = 0,060 + 0,307 = 0,367 



998 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 492. 



also Polytechn. Centralblatt, 1847). Fig. 836 gives an ideal representation 

of the tachometer in a pipe. The vane-wheel in 
this case also puts a shaft B E in rotation by 
means of an endless screw ; the former passes out 
of the pipe B i?, in which the water to be 
measured flows, through a stufRng-box F into 
the case O jffof the counting apparatus, the ar- 
rangement of which may be very varied. 

Remark — 3) The French have but lately be- 
gun to give sufficient attention to the tachometer. 
A complete treatise upon this instrument, by 
Baumgarten, is to be found in the " Annales des 
ponts et chaussees," T. XIV, 1847, and an abstract 
of it in the "Polytechnisches Centralblatt, 1849." Baumgarten recommends 
a screw-wheel and adds several remarks, which agree very well with our 
experiments, made many years ago. A new tachometer, without wheels 
and with a long screw, is described by Boileau in his " Traite de la mesure 
des eaux courantes." 




Fig. 837. 



§ 492. Pitot's Tube. — The other hydrometers are more im- 
perfect than the tachometer ; for they are either less accurate or 
more difficult to use. The simplest instrument of this kind is 
Pitot's tube (Fr. la tube de Pitot; Ger. Pitot'sche Eohre). It con- 
sists of a bent glass tube ABC, Fig. 837, which is held in the 

water in such a manner that the lower part is 
horizontal and opposite to the motion of the 
water. By the impulse of the water a column 
of water will be forced into the tube and held 
above the level of the water, and this rise I) B 
is proportional to the impulse or to the velo- 
city of the water which produces it ; this rise 
or difference of level can therefore serve to 
measure the velocity of the water. If the height 
I) E above the exterior surface of the water 
= Ji and the Telocity of the water := v, we can put 




A = 



V' 



2g fj. 



2? 



in which fi is an empirical number, or inversely 

V = fj^ ^2 g h, or more simply 



§ 493.] 



HYDROMETRY, ETC. 



999 



In order to find the constant ip, we hold the instrument in the 
■water where the velocity is known to be Vi ; if the rise is — h^, we 



Fig. 838. 



have the constant ijj — ^ , which can be employed in other cases, 

where the velocity is to be determined by this instrument. 

In order to facilitate the reading ofi" of the height h, the instru- 
ment is composed of two tubes A B and G D, as is represented in 
Fig. 83xS ; from one of the tubes a pipe proceeds in the direction of 
the stream, and from the other two pipes F and F^ at right-angles 
to that direction, but by means of the same cock 
both tubes can be closed at once. If we draw the 
instrument out of the water, we can easily read off 
the difference of height K L = li of the columns 
of water upon the scale placed between them. In 
order to prevent the water from oscillating in the 
tubes, it is necessary to make their mouths narrow ; 
and in order that the cock may be shut quickly and 
certainly, it is provided with a crank and a rod 
H S, which is represented in the figure principally 
bv a dotted line and terminates near the handle of 
the instrument. 

Remark — 1) Although Pitot'stube is not so accurate 
as the tachometer, yet, on account of its simphcity, it 
can be highly recommended. The author has discussed 

L this instrument at length in the " Polytechnisches Cen- 
tralblatt, 1847," and gives there a series of numbers, de- 
termined by exf)eriment, and the values of the coefficient 
■0 deduced from them. With fine instruments, when the 

[r velocities were between 0,32 to 1,24 meters, we found 

'D = 8,545 \/h meters. 

3) Duchemin recommends Pitot's tube with a float. 
Since the latter must be pretty wide, it dams the water 
back to a certain extent, so that it cannot be employed for narrow canals 
(see Duchemin : " Recherches experim. sur les lois de la resistance des 
fluides"). Boileau describes in his work, cited in § 412, a new kind of 
Pitot's tube, which is provided with a small gauged vessel ; the velocity 
is measured by the quantity of water pressed above the surface of the 
water. 



F 



§ 493. Hydrometric Pendulum. — The liydrometric pendu- 
lum (Fr. pendule hydrometriquj ; Ger. Stromquadrant or hydro- 



1000 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 493. 



Fig. 839. 





c 


r-5 


inHiTl 






mxi 




A 









■- ' - 


^— \= 







metrisches Pendel) was principally employed by Ximenes, Michelotti, 
Gerstner, and Eytelwein to measure the velocity of running water. 

This instrument consists of a quadrant 
A B, Fig. 839, divided into degrees and 
parts of a degree, and of a string attached 
to its centre C, at the other end of which is 
fastened a metal or ivory ball K, 2 or 3 
inches in diameter. The velocity of the 
water is given by the angle ACE formed 
by the stretched string with the vertical, 
when the plane of the instrument is placed 
in the direction of the stream, and the 
ball is immersed in the water. Since the angle cannot easily exceed 
40°, this instrument often has the form of a right-angled triangle, 
and the graduation is then marked upon the base. In order to 
place the zero line vertical, we can either place a level upon the 
mstrument or we can employ the ball itself by allowing it to hang 
out of the water and then turning the instrument until the string 
corresponds with the zero line. For velocities less than 4 feet we 
can employ an ivory ball ; for greater velocities, however, we must 
use heavy balls of metal. On account of the vibrations of the ball, 
not only in the direction of the motion of the water but also in 
that at right angles to it, it is always difficult to read off the angle, 
and the result is never free from uncertainty; this instrument 
cannot therefore be considered to be a perfect one. 

The dependence of the angle of deviation, for a ball that is not 
deeply immersed, upon the velocity of the water can be determined 
in the following manner. The weight G of the ball and the im- 
pulse of the water P = fi F v^, which increases with the cross- 
section F of the ball and the square of the velocity v, give rise to 
a resultant R, which is counteracted by the string and is deter- 
mined by the angle of deviation (5, for which we have 



tang, 6 = -— = — ^- 



or inversely 



V' 



G tang. 6 



and 



a/ ^ 
^ fjbF 



Vtang, 6, 



X.£i«, 



V = ifj Vtang. 6, 
in which '^ is an empirical coefficient, which must be determined 



§494 



HYDROMETRi, ET 



o. 



1001 



in the manner stated above (§ 491) before the instrument can be 
used. 

§ 494. Rheometer. — The remaining hydrometers, such as 
Lorgna's water-lever, Ximenes' water-vane, Michelotti's hydrauhc 
balance, Brunning's tachometer and Poletti's rheometer, etc., are 
difiBcult to use and partially uncertain. The principle of all of 
them is the same; they consist of a balance and of a surface, 
which is subjected to the impact of the water ; the former serves 
to measure the impulse P of the water against the former, but 
since the impulse is = fi F v^, we have inversely 



v=V 



p 

\i F 



= ipVP, 



in which ip is an empirical constant, dependent upon the magni- 
tude of the surface subjected to the impulse of the water. 

The Eheometer, which has been lately proposed by Poletti, does 
not differ essentially from Michelotti's balance and consists of a 
lever A B, Fig. 840, movable about a fixed axis G, and of a second 

arm C D, upon which a surface, or, according to 
Poletti, a simple rod, which is to be subjected 
to the impact, is screwed. In order to balance 
the force of impact of the water, shot or weights 
are put into the sheet iron box, which is sus- 
pended at A upon the lever, and to balance 
the empty apparatus in still water, weights 
are hung at B, the extreme end of the arm 
C B. From the weights added at G and the 
arms of the lever G A — a and G F = b, we 
obtain by means of the formula P b = G a the 




impulse 



a 



P = T G and v 



/A=/ 



fi F 



a G 
mb F 



'ipVG, 



in which i/^ denotes an empirical constant. 

A hydrometer constructed upon the same principle, in which 
the impulse of the water is balanced by the force of a spring (hy- 
drometre dynamometrique) is described by Boileau in his treatise 
upon the measurement of water. 

Remabk 1. — The last-mentioned hydrometers are discussed at length in 

Eytelwein's " Handbuch der Mechanik," Vol. II, in Brunning's " Abhand- 
lung iiber die Geschwindigkeit des fliessenden Wassers," in Venturoli's 



1002 



GENERAL PRINCIPLES OF MECHANICS. 



[§495. 



"Element! di Meccanica e d'Idraulica," Vol. II. Concerning Poletti's 
Rheometer, see Dingler's Polytechn. Journal, Vol. XX, 1826. Stevenson's 
hydrometer is Woltmann's tachometer, see Dingler's Journal, Vol. LXV, 
1842. The water-meters and gas-meters constructed like reaction wheels 
will be treated in the following chapter. 

Remark 2. — A work to be particularly recommended for practical 
purposes is the " Hydrometrie oder practische Anleitung zum Wasser- 
messen von Bomemann, Freiberg, 1849." Boileau's work has already been 
mentioned several times (see § 412, etc.). 



w 



CHAPTER IX. 



OF THE IMPULSE AND RESISTANCE OF FLUIDS. 



Fig. 841. 



§ 495. Reaction of Water. — The total pressure of the still wa- 
ter in a vessel is, according to § 362, reduced to a vertical force equal 
to the weight of the mass of water ; but if the vessel A F, Fig. 841, 

has an opening F, through which 
the tirvater issues, this force under- 
goes a change not only because a 
portion of the wall of the vessel is 
absent, but also because the water, 
which issues from the orifice, like 
every other body, which changes 
its conditions of motion, reacts by 
virtue of its inertia. The change 
in the motion of a body may consist 
either of a change of velocity, or 
of a change of direction, and, there- 
fore, the reaction (Fr. reaction ; Ger. Eeaction) of the issuing water 
may be due not only to an acceleration but also to a constant 
change in the direction of the water, which is approaching the 
orifice. 

We can make ourselves acquainted with the complete reaction 
of the water in a discharging vessel in the following manner. 

Let c be the velocity of the water, which is issuing from the 
orifice F, d the relative velocity of the water at the surface A, 




§495.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1003 

G the area of this surface and h the head of water A D Sbt the ori- 
fice. Then we have 

2g 2g' 

and the discharge 

Q = Fc = Gc„ 
If we imagine the vase A F, Fig. 841, to move forward in a 
horizontal direction with a velocity v, we must put for the absolute 
velocity Ci of the water entering the vessel 

and if the angle of inclination of the axis of the stream to th^ 
horizon i^ E F c — a, we have for the absolute velocity w of the 
effluent stream 

w"^ = c^ + v"^ — '^ c V COS. a, 
Now the actual energy of the water before efflux is 

and that after efflux it is 

r ^"^ ^ /c" + v" — 'H c V cos, a\ ^ 
^^=¥-g^^ = [ ¥-y )'?''' 

hence the energy withdrawn from the water and transmitted to 

the vessels is 

-^ ^ T- (Ci^ — c' + 2 c V cos. a , \ ^ 
L = L, - L,= (^ — + hj Qy, 



& Ci 



or, since r ^— = li, 

^g ^g 



-r c V COS. a _ 
L — Q y. 

9 

The horizontal component of the reaction of the water is 

^^ L ' c COS. a „ 
V g ^ ^ 

Since Q — F c,yfQ have also 

H = — F y COS. a = 2 . r-~ F y cos. a = % h F y cos. a, 
g 2 (/ 

and therefore, when the direction of the stream is horizontal, as in 
Fig. 842, 

H=%hFy. 

Therefore, tM reaction of a horizontal stream is equal to ths 
weight of a column of water, whose cross-section is that of the stream 
and whose height is double that (2 h) due to the velocity. 



1004 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 496. 




Remark. — Mr. Peter Ewart, an Englishman, has recently made experi- 
ments to prove the correctness of this law (see " Memoirs of the Manchester 
Philosophical Society," Vol. II, or the " Ingenieur, Zeitschrift fur das ge- 

sammte Ingenieurwesen," Vol. I). He hung 
the vessel H B F ujpon. a horizontal axis (7, 
Fig. 842, and measured the reaction by a bent 
lever ABB, upon which the vessel acted by 
means of a horizontal rod A G, which pressed 
against the vessel exactly opposite to the ori- 
fice F. For efflux through an orifice in a thin 
plate, he found 

P=l,U.~Fy. 

2g 

If we put the cross-section 
F^ =z 0,64 F 
and the efiective velocity of discharge 
v^ = 0,96 V 
(see § 405), we obtain by the theoretical formula 

2 2 2 

P=2.'^ .F^y = 2. 0,96^ . 0,64 .^Fy = 1,18 ^ i^ r, 
2 g ^ 2g ' - 2g '^ 

or about the same that was given by experiment. With an orifice shaped 

like the contracted stream, he found P — 1,73 -^r — F y. and the coefficient 

2 g 

of efflux or velocity = 0,94. Since in this case F^ = i^and v^ = 0,94 o, 
we have theoretically 

2 g ' ^ 2 g '^ 

which agrees very well with the result of the experiment. 

§ 496. If we imagine the discharging vessel A F, Fig. 843, to 
be moyed vertically upwards with a velocity v, we have for the 

absolute velocity of the water which 
enters it 

and, on the contrary, for that of the 
water issuing from it (the same no- 
tations being employed as in the 
foregoing paragraph) 

w"" = 6" + v"" + 2 cv COS. (90° -}- a) 
= c^ + v^ — 2 c V sin. a. 
Hence the total energy of the 

volume of water Q per second is 

_l(v-cf 



Fig. 843. 




. = (! 



^g 



+ h 



)Qy, 



and, on the contrary, that of the water discharged is 
Li ~ {c^ -r- v'^ — 2 c V sin. a) Q :2 g 



§ 496.] 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1005 



consequently the mechanical effect imparted by the water to the 
vessel is 

Ci -\- Ci^ — c^ -\- 2 c V sin. a 



L = Lr 



- (ll. 



■J C Ci 

or, smce h — ^ ^r— , 



(c sin. a — Ci) V 



+ h 



)Qy> 



Qy, 



and the corresponding vertical force is 

c sin. a — cA ^ / . 

— I Q y ■= \sin. 



r=^ = ( 



V 



9 



a 



F\ c_ 
g) g 



^)^^r 



/ F\ c" I F\ 

= \si7i. a — —I — F y =[sin. a — —I .2hFy. 



9 
If the orifice of efflux is small, compared to the surface G, we 

F 

have -^ = 0, and, therefore, the vertical component of the reaction 

V — 2 h Fy sin. a. 
According to the foregoing paragraph the horizontal compo- 
nent of this force was 

H = 2 h F y COS. a ; 

hence the total reaction of the water is 

E = VV + H' = 2hFy, 
and its direction is exactly opposite to that of the motion of the 
effluent water. 

\i F = G, I.E., if the water flows through a pipe of uniform 

F 
width, we have -^ = 1, and therefore 

V = (sin. a — l).2hFy= — (1 — sin. a) .2 h Fy; 
in this case V does not act upwards but downwards, and the total 
reaction is 

R = VV + H' = Vcos. a' + (1 - sin. ay .2hFy 

Fig. 844. = V2 (1 — sin. a) .2h Fy 

= 4:hFysin. (45° - |). 

For a — — 90°, i.e., when the pipe forms k 
semicircle, E = 4= h Fy ' "~t' — .__.:. 

If a — -\- 90°, we have the case represented in 
Fig. 844, where IT = and 




1000 



GENERAL PRINCIPLES OF MECHANICS. 



[§49a 



F 

consequently, for — = 0, we have 

The total weight of the water m the vessel will be diminished 
that much, when the water is allowed to -flow out. 

§ 497. Impulse and Resistance cf Water. — Water or any 
other fluid, when it impinges upon a solid body, imparts a force or 
impulse to it, and thus produces a change in its state of motion. 
The resistance (Fr. resistance ] Ger. Widerstand), which water 
makes to the motion of a body, is not essentially different from im- 
pulse. The examination of these two forces constitutes the third 
chief division of hydraulics. We distinguish from each other first, 
the impact of an isolated stream (Fr. choc d'une veine de fluide ; 
Ger. Stoss isolirter Wasserstrahlen) ; secondly, the impact of a 
hounded stream (Fr. choc d'un fluide defini; Ger. Stoss im be- 
grenzten Wasser oder Gerinne) ; and thirdly, the impact of an unlim- 
ited stream (F. choc d'un fluide indefini ; Ger. Stoss im unbegrenz- 
ten Wasser). Impact of the first sort takes place when a stream 
discharged from a vessel encounters a body, as, E.G., the bucket of 
an overshot water-wheel ; impact of the second sort occurs, when 
the water in a canal or trough strikes against a body which en- 
tirely fills the cross-section of the latter, as, e.g., the float of an 
under-shot water-wheel. Finally, impact of the third kind occurs, 
when running water strikes upon a body immersed in it and the 
cross-section of the latter is but a small part of that of the stream, 
as, E.G., the float of a wheel in an open current. 

We distinguish also impact against bodies at rest and todies in 

motion, against curved and plane 
surfaces ; the latter may be either 
direct or oblique. 

We will now consider a more 
general case, viz. the impact of an 
isolated stream against a surface 
of revolution, moving in the direc- 
tion of the motion of the stream, 
which coincides with the direction 
of the axis of the surface. 

§ 498. Impact of an Isolated 
Stream.— Let B A B, Fig. 845, be 



Fig. 845. 




§498.] THE IMPULSE AND RESISTANCE OF FLUIDS. 100? 

a surface of revolution, A P its axis, and F A q> stream of water 
moving in the direction of the axis of the latter and impinging 
against it; let us put the velocity of the water = c, that of the 
surface = v, and the angle B T P, which the tangent D T to the 
end B of the generatrix or each fibre B I) of the stream of water, 
which leaves the surface, makes with the direction B U of the axis, 
= a, and let us assume that tlie water does not lose any vis viva in 
consequence of the friction while passing over the curved surface. 
The water impinges upon the surface with the velocity c — v and 
then passes over the surface with that velocity and leaves it in a 
tangential direction T B, T B, etc., with the same velocity. From 
the tangential velocity B D = c — v and from the velocity B E 
= ?; in the direction of the axis, we obtain the absolute velocity 
B C — c^ of the water, after it has impinged upon the surface, by 
the well-known formula 



c^ = V(c — vY -f 2 (c — v) V COS. a + v\ 
Now a discharge Q can produce by its vis viva a mechanical 

effect ^— , Q y, when it loses its entire velocity c ; hence the energy 

c ^ 
remaining in the water is = ^ . Q y, that transmitted to the sur- 

face is 

Pv = 4- Qy - -^ Qy = ^^-^^ -Qy 
^ g ^ g ^ g 

_ {/f — (c — vY — 2 {c — v)v cos. a — v"^] ^ 

- 2^ yy 

_ 2cv — 2v'^ — 2(c — v) V cos. a 



^g 

P V =, {\ — COS. a) '— Q y, 



Q y, I.E. 



and the/orce or impulse in the direction of the axis is 

P = (1 — COS. a) Q y. 

If the surface moves with a velocity v, which is in the opposite 
direction to that of the water, we will have 

P = (1 - COS. a) fci-^ Q y, 

9 
and if the surface does not move oy \iv = 0, the impulse or hydrau- 
lic pressure in the direction of the axis is 

c 
P = (1 — COS. a) - * Q y* 



1008 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 499. 



From this it follows that the impulse of one and the same 7nass 
of water, ivhen the other circumstances are the same, is proportional 
to the relative velocity c ^ v of the ivater. 

If the area of the cross-section of the stream is F, the volume 
of the impinging water is F {c =f v) ; hence 

p = (i 

or for V = 0, 



cos. a) ^ '- Fy ; 



P = (1 - COS. a)~ Fy. 

g 

If the cross-section of the stream remains the same, the impulse 
against a surface at rest increases with the square of the velocity of 
the ivater. 

§ 499. Impact against Plane Surfaces. — The impulse of 
the same stream of water depends principally upon the angle a, at 
which the water moves off from the axis after the impact; it is 
null when this angle = 0, and, on the contrary, a maximum and 

{c qp. v) 



2 



g 



Qy 




Fig. 847. 



when this angle is 180° or when its cosine = — 1, in which case, as is 

represented in Fig. 846, the 
water quits the surface in a di- 
rection opposite to that in which 
it struck it. In general the im- 
pact is greater against concave 
than against convex surfaces; 
for in the former case the angle 
is obtuse and its cosine negative 

and 1 — COS. a becomes 1 + cos. a. 

Usually the surface is, as is represented in Fig. 847, plane and 

therefore a = 90° or cos. a = and the impulse 

P^i^^^.Qy. 

g 

When the surface is at rest, we have 

n O 




' ^y-^-i-g^y 



%Fhy. 



P = -Qy = 

g^^ g 

The normal impulse of water against a plane surface is equal to 
the weight of a column of water, the cross-section of whose base is 
equal to the cross-section of the stream, and whose height is twice 

that due to the velocity ( 2 7i = 2 . x- ). 



§ 499;^ 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1009 



Fig. 848. 




The results of the experiments made upon this subject by 
Michelotti, Vince, Laugsdorf, Bossut, Morosi and Bidone were 
about the same, when the cross-section of the impinged surface was 

at least 6 times that of the stream 
and when this surface was at a 
distance not less than twice the 
thickness of the stream from the 
orifice. The apparatus employed 
consisted of a lever like Poletti's 
Eheometer (§ 494), upon one end 
of which the stream impinged, 
the impulse was balanced at the 
other end by weights. The ap- 
paratus employed by Bidone is 
represented, in Fig. 848. B C is the surface subjected to the action 
of the stream, G the scale-pan for receiving the weights, D the axis 
of rotation, and K and L are counter weights. 

Remark. — The most extensive experiments u^jon the impulse of water 
were made by Bidone (see " Memoire de la Reale Accademia delle Scienze 
di Torino," T. XL, 1838). They were made with a velocity of at least 27 
feet and with brass plates of from 2 to 9 inches in diameter. Bidone gen- 
erally found the normal impulse against a plane surface somewhat greater 
than 2 Fh jy but this increase is to be ascribed to the increase of the arm 
of the lever, in consequence of the falling back of the water. See Duchemin : 
Recherches experimentales sur les lois de la resistance des fluides (translated 
into German by Schnuse). When the impinged surface was very near the 
orifice, Bidone found P to be only 1,5 F h y. When the impinged surface 
was of the same size as the stream, in which case the angle of deviation a 
is acute, according to du Buat and Langsdorf, P is only = Fhy. Bidone 
and others have found that the impulse during the first instant was nearly 
twice the permanent impulse. Comparative experiments upon the impulse 
and reaction of water hav^e been made by the author with a reaction wheel. 
See his " Experimentalhydraulik" and the " Civilingenieur," Vol. I, 1854. 

By more recent experiments upon the impact of isolated streams of air 
and water (see Civilingenieur,Vol. VII, No. 5, and Vol. VIII, No. 1), the 
author found the effective impulse of an isolated stream of air or water 
against a normal plane to be 92 to 96 per cent, of the theoretical force P = 

—^^ that, on the contrary, the impulse of such a stream against a hollow 

surface of rotation by which the direction of the stream is made to deviate 
an angle 6 = 134°, is but 83 to 88 per cent, of the theoretical force P = 

c(l - eo8. 6) ^-^. 
9 

64 



1010 GENERAL PRINCIPLES OF MECHANICS. [§ 500. 

§ 500. Maximum Work done by the Impulse. — The me- 
chanical effect 

Fv= (1 - COS. a) ^^ ~ ^^ ^ Qy 

9 
depends principally upon the velocity v of the impinged surface ; 

E.G. it is null not only for v — c, but also for ?; = 0; hence it fol- 
lows that there must be a velocity, for which the work done by the 
impulse is a maximum. It is evident that this is the case when 
\c — v) v\^2^ maximum. If we consider c to be half the periphery 
of a rectangle and v to be its base, we have its height — c — v and 
its area — (<? — v) v ; now the square is that rectangle, which has 
the greatest area for a given periphery ; hence (c — v)v\&2. maxi- 

mum, when {c — v) = v, I.E., v =-, and we obtain the maximum 

mechanical effect of the impulse, when the surface moves in the 
direction of the stream with half the velocity of the latter ; the 
work done is then 

P V = {1 — COS. a) . }y . — . Q y = (1 — cos. a) , ^ Q Tiy. 

Zg 

Now if a =: 180°, I.E., if the motion of the water is reversed by 
the impact, we have the work done 

= %.\Qhy =^ QJiyy 
but if a = 90°, I.E., if the stream strikes against a plane surface, 
the work done is but ]- QJiy,\n this case the water transmits to 
the surface but one-half of its actual energy, or but one-half of the 
mechanical effect corresponding to its vis viva. 

Example — 1) If a stream of water, the area of whose cross-section is 40 
square inches, deHvers 5 cubic feet per second and strikes normally against 
a plane surface, which moves away with a velocity of 12 feet, the impulse is 

P = ^J^ Qy = (-^^ - is) . 0,031 . 5 . 62,5 = 6 . 0,031 . 312.5 

= 58,125 pounds, 
and the mechanical effect transmitted to the surface is 

Pv = 58,125 . 12 = 697,5 foot-pounds. 
The maximum effect is obtained, when 

c 5 . 144 ^ ^ , 

" = 2 = ^-To"-^^^^'*' 

and it is 

i = ^,^.^7=^.18'^. 0,0155 . 5 . 62|^ = 81 . 0,155 . 63,5 = 784,6875 

foot-pounds ; 
the corresponding impulse or hydraulic pressure is 

_ 784,6875 



9 



= 87,19 pounds. 



501.] THE IMPULSE AND RESISTANCE OF FLUIDS. 



1011 



2) If a stream F A^ Fig. 849, the area of whose cross-section is 64 square 
inches, impinges with a velocity of 40 feet upon an immovable cone, 

whose angle of convergence B A B = 100°, the 
hydraulic pressure in the direction of the stream is 



Fig. 849. 




P = (1 — COS. a) - Q y 



64 




= (1 - COS. 50°) . 40 . 0,031 . --^ . 40 . 63,5 

144 ' 

= (1 - 0,64379) . 1,24 . "^'^ 

= 0, 35721 . 1377,8 = 492,16 pounds. 

§ 501. Impact of a Bounded and of an Unlimited 
Stream. — If we surround the periphery of a plane surface B B, 
Pig. 850, with borders B D, B D (Fr. rebords ; Ger. Leisten), which 
project beyond the surface struck by the water, the 
water will be deviated from its course at an obtuse 
angle as in the case of concave surfaces, and the 
impulse is greater than when the surface is plane. 
The action of this impact depends principally upon 
the height of the border and upon the ratio of the 
cross-section of the stream to that of the enclosed 
surface. In an experiment, where the stream was 
one inch thick and the cylindrical border 3 inches in diameter and 
3^ lines high, the water flowed from the surface in nearly the oppo- 
site direction and the impulse was 

3,93 ^i^^r; 

in all other cases this force was smaller. It is impossible ever to 

attain the theoretical maximum value 4 — i^ y in consequence 

of the friction of the water upon the surface and upon the border. 
In the case of the impact of the bounded stream FAB, Fig. 

851, there is also a border ; it is, however, only partial and includes 

but a portion of the periphery; it 
limits, moreover, both the stream and 
the impinged surface. The imping- 
ing stream is turned in the direction 
of the portion of the periphery, which 
has no border, and is therefore de- 
viated 90° from its original direction ; 



Fig. 851. 




hence the formula, which we found for the isolated stream, 



1012 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 50». 



P = 



_{c - v) 



Qy 



f-^-) 



cFy, 



9 ^ (J 

holds good here. If the surface B B, Fig. 847, against which the 

stream strikes, moves away with a velocity v in a direction, which 
forms an angle (5 with the original direction of the stream, the ve- 
locity of this surface in the direction of the impact is 

^1 = -y COS. d ; 
hence the impulse is 

p ^ (g - ^ co s, (?) ^ ^ 

and the work done by it per second is 



L^P v,= 



__ {c — V COS. d) V COS. d 



9 



Qr 



The principal application of this formula is to the impact of an 
unlimited stream, in which case 

Q = F {c — V COS. (5), and therefore 

[C — V COS. 6)'' 



P = 



9 



Fy. 



§ 502. Oblique Impact. — There are several cases of oblique 
impact, viz. : where the water after impact flows away in one, in 
two or in more directions. If, as in the case of the impact of a 
bounded stream, the surface A B, Fig. 852, has a border upon 
three sides so that the water can flow away in one direction only, 
we have the hydraulic pressure of the water against the surface in 
the direction of the stream 

P — (1 — COS. a) 

Fig. 852. 




••mmimf"irr<'"n'"""m'i'ifr'm,-r,-',-,„„„„„„„„„„„.,„„„„„ 




But if the Impinged plane B (7, Fig. 853, has a border upon 
two opposite sides only, the stream divides itself into two unequal 
parts, the angle of deviation a of the larger part ^i is less than 
that 180° — a of the smaller part §2 and the total impulse in the 
direction of the stream is 

P = (1 — COS. a) . — - — gi y + (1 + COS. a) . — - — ft 7 



9 



9 



§503.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1013 
1 [(1 — COS, a) §1 + (1 + COS. a) Q.^ y. 

y 

But the conditions of equilibrium of the two portions of the 

stream require that the pressures 

(p — tjt \C ""~ ffS 

^ (1 — COS. a) §1 y and ^^ ^ (1 + cos. a) Q^ y 

y y 

shall be equal to each other ; hence 

(1 — COS. a) ^1 = (1 + COS. a) Q^, 
or, since Q = Q^ + Q<^, we can put 

(1 — COS. a) ^1 =: (1 + COS. a) (Q — Q^), i.e. 

Q, = (1-±|^«) Q and Q, = (1^:|?^) Q, 

so that the total impulse in the direction of the stream is 

P = (g^i;1.2(l-cos.a) <^+7-")^ y 
g '2 

= -^ (1 - COS. a) Q y, i.e. 

Dividing the work done by the impulse in a second 

T D {^ — '^) -2 r^ 

L = F V = ~ V sm. a . Qy 

. 9 ^ . 

by the velocity A Vx~ Vi^=^ v sin. a, with which the surface recedes 
in a normal direction, we obtain the normal impulse 
,^ (c — v)v sin.'' a ^ (c — v) . ^ 

. g V sm. a g 

which consists of the idaralM impulse 

P= JSfsin. a = ^ ^ sin."^ o, > Qj, 

g 

«nd of a lateral impulse 

o liT (c — v). ^ c — v,^ „ 

b = JS COS. a = '- sm. a cos. a . Qy = — — sm. 2 a Qy. 

The normal impulse is proportional to the sine, the parallel im- 
pulse to the square of the sine of angle of incidence, and the lateral 
impulse to the sine of double this angle. 

If, finally, the oblique surface, which is struck, has no border, 
the water can flow away in all directions and the impulse is still 
greater ; for a is the smallest angle which the fibres of water can 
make with the axis ; hence every fibre which does not move in the 
normal plane exerts a greater pressure than those which do. If we 
assume that the angles of deviation of one portion §„ which corre- 



1014 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 503. 



Fig. 854. 



spends to the sectors A B and DOE, Fig. 854, are C -F = a 
and COO — 180° — a, that those of the other portion ^o? which cor- 
responds to the sectors A E 
and B 0D,2.vqG OK=COH 
— 90°, and that the two por- 
tions produce equal parallel im- 
pulses, we can put 

Qx y sm. a 




P = 



+ 



C — 7) 



ft% 



and, since Qx sin.'' a — Q^ and 
Q = Q\ -\- Qi, it follows that 

^1 (1 + sin.'' a) — Q, 



and that the total joarallel impulse is 

v\ 2 § y sin.^ a 2 sin.'' a c — v 

9 



P = 



f-i-') 't 



■ Qr- 



+ sin.'' a 1 + sin." a ' g 
Although this assumption is only approximatively correct, yet 
the results of the latest experiments by Bidone agree very well 
with it. 

Remakk. — Prof. Brock, in his Mechanics, page 614, finds for oblique 
impact against a circular surface 

P = ( ^ — a J tang, a C ) Q 7, and 

-, o. I<^ — '^\ ^ 

W = tang, a I. cotg. -^ \ j Q y. 

§ 503. Impact of Water in Water. — If a certain quantity 

Q of water discharges with a velocity A c = c into a vessel D E, 
Fig. 855, which is moving with a velocity A v = v, a part only 



Fig. 855. 




Lx — -|-- y of its actual energy L^ 



Qc" 

^9 



y will be expended in producing 



and maintaining the eddy A B, 
which is due to the loss of velocity 
Ci. If we denote by a the angle v A c, 
made by the direction of the stream 
with that of the motion of the ves- 
sel, w^e have 
Ci = c' -\- v'' — 2 cv COS. a, 



§504.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1015 

and, therefore, the mechanical effect lost in consequence of the 

eddy j _ 6 (^' + ^'' — ^ ^ ^ ^os. a) 

l^x — . 

2 g 

As the volume Q of water participates in the motion of the vessel, 

its velocity v is the same as that of the latter, and the energy, 

Q v" 
which it still possesses, is X2 = -^ — y ; hence the energy which is 

transmitted to the vessel and expended in moving it forward, is 

L ^^ Lq — L\ — Li 

/c' — {c^ + v'' — ''Z c V COS. a) — v\ ^ _ 2cv cos. a — 2v^ 



\ ^ zcvcos. a — 2v ^ 

-)Qy = ^, Qr 



\ 2g / ^' 2g 

(ccos.a — v)v ^ 

-, — ^^' 

and the force with which the vessel is urged forward in the direc- 
tion of its motion by the water which flows into it is 

^ L Ic COS. a, — v\ ^ 

^ = v = { — J—) ^ >'• 

Now the discharge per second, which impinges against the 

vessel, is Q = F c, F denoting the cross-section of the stream at 

its entrance ; hence we have 

-^ (c COS. a — v) c „ 
F — '— F y, 

fj 
and for the case when the vessel is at rest, or when v = 0^ 

2 2 

P = '— Fy = 2 77— F y cos. a — 2 Fliy cos. a, 

g ' 2g ' 

in which li denotes the height -— due to the velocity. 

The mechanical effect is a maximum for ?' = i c cos. a and it is 
(^ cos ^ d 

L,n — \ — w^ — Qy'=^\Q'J^y cos? «• 

If the direction of the stream is the same as that of the motion 

of the vessel, a = 0, and we have 

{c — v)v 

L = — Q y and 

9 

In this case but half the total energy Q h y oi the water is utilized 
(compare § 500). 

§ 504. Experiments "with Reaction Wheels. — Tlie best 
method of proving the above tlieory of the impact and reaction of 
water is to make use of a reaction wheel A A //, Fig. 856, with a 



1016 



GENERAL PRINCIPLES OP MECHANICS. 



[§ 504. 



yertical axis of rotation O D (see the author's " Experiraental-Hy- 
draulik/' § 48, etc.). The water which turns the machine enters 
into the receiver A A of the wheel nearly tangentially through two 



Fig. 8 




lateral canals E, E, and is discharged through two lateral orifices 
F, F in the ends of the reyolving tubes R, R. In order to maintain 
the efflux of water constant and the rotating force inyariable, the 
pipe which conveys the water to the reservoir G is provided with a 
cock II ; from the reservoir the water is conveyed by the pipe XL 
to the chamber E E, into which the canals E, E open. While the 
machine is in operation, the cock II must be turned in such a 
manner that the surface of the water in the reservoir G shall 
always touch the end of the pointer Z. 

"When we wish to determine the reaction of the effluent water, 



§505.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1017 

a thin string S, to one end of which a weight is attached, is passed 
over a pulley and then wrapped round the central tube E. The 
quantity of water discharged is measured in the reservoir, from 
which the water flows into the pipe with the cock IT, by observing 
the area A of the surface of the water and the distance a which it 
sinks during the experiment. If the duration of the observation is 
= t, we have the discharge per second 

and if the fall, i.e. the vertical distance between the surface of the 
water in the reservoir G and the orifice of discharge of the wheel 
= h, the total energy of the water discharged per second is 

T -. , A ahy 

L = Qhy = — ^. 

Now if the machine has raised the weight G a distance s in the 
time t, the work really done by the wheel m a second is 

and we can now compare these two values, the second of which is 
always the smaller. 

§ 505. Theory of the Reaction Wheel.— The total fall h 
in such a wheel consists of the fall h^ from the surface of the water 
to the pomt E, where the water enters the wheel, and of the fall 
^2 from the latter point to the orifice, by which the water leaves the, 

wheel. From hi we calculate, by means of the formula Ci = V2 g Aj, 
the velocity with which the water enters the wheel, and from h^, 
according to § 304, by means of the formula 

c =z V2 ff h^ -^ v"" — v^^ 
the velocity with which it quits it, when the velocities of rotation 
Vi and V of the wheel at the points of entrance and exit are known. 
Since the direction of this reaction of the water, which acts as the 
rotating force, is opposite to that of the velocity of discharge, the 
absolute velocity of the water upon leaving the wheel is 

w = c — V, 
and its square 

to'' = c^ — 2cv ^- v"" = %gh,-2cv -\- 2v^ - v,^\ 



1018 GENERAL PRINCIPLES OF MECHANICS. [§505. 

hence the energy of the effluent water is 

U-^Qy,^^ = Qy[h, -g %g)- 

The water, which enters the wheel with the relative velocity 
Wx ^=^ Cx — Vi, loses (according to § 436) by the impact the energy 

and consequently of the total energy 

Qhy = Q {hx + h,)y, 
only the portion 

L=Qy(h-lix-lh) + Qy\^- — j-^^—-'j = Qy\ ^ ^ + — - j 

is transmitted to the wheel. 

In order to obtain the greatest amount of work from the wheel 
we must have w = or v = c and iVx = or ?;i = Cx, and therefore 

— L = h^ or Vx ~ V2 ff hi, as well as 

-^ :=z Jix or Vx = "^2 a hi. 

In this case, therefore, lix = Jh = i ^i' and the corresponding 
maximum effect of the machine is 

I.E., equal to the total energy of the water. 

If Tx denotes the distance of the point of entrance and r that of 
the orifice of exit of the wheel from the axis, we have 

Vx Tx , Tx 

— = — ■, whence Vx~ — v, 
V r T 

and, m general, the rate of work of the wheel 

(7' \ 1) 

so that the rotating force, measured at the distance r, is 

(^C-V + ^Cx). 



P ^^=Q y ( ^ .. ,^1 



'^9 

If the arm of the suspended weight or load is a, which in the ap- 
paratus represented is very nearly the radius of the central tube B, 
we have G a — P r, and, therefore, the weight to be attached and 
to be raised during the rotation of the wheel is 

6^ = - P = — ^ Uc - v)r + Cx rx\ 
a g a 

or for c =^ V and Cx = Vx^ 



§505.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1019 

ga g a 

If F denote the area of the orifices of eflSux and F^ that of liose 
of influx, we have 

Q — F c = Fi c^, and therefore 

i-. = -^-^==and 

/ 



F = ^ = -,^=J=^=F.V\ ,^^4 



^ V)lg U, + v' - i\^ ^gh, ^- V- - v,^ 

For V = c and i\ = c^, in which case hi z= h^ = ^ 7^, we have 
Q = F V, and therefore 

V 



on the contrary, for ?; = 0, Q = F V2 g h.^, and therefore 

g \ r / 

If we allow the water to enter the wheel slowly, we can put 
Ci = and h^ = and the force of the reaction in the last case 

becomes 

^ Fc'r 2 Fc'r _ ^, ^ ^, 

F = = —^ — 2Fh^y — 2Fhy, 

g ^g ^ ^' 

as we found above. 

Since in these calculations we neglected the passive resistances, 
the experiments with the machine represented do not give the 
values for the force found above, but values which are a few per 
cent. less. However, the results of experiments carefully made 
with such a wheel agree very well with the theory just demonstrated. 

"When we wish to make use of this machine to test the theory 
of the impact of ivater, we begin by removing the chamber F F &o 
as to allow the water to enter near the centre without any velocity 
of rotation, and we then fasten opposite to the orifices m the re- 
volving tubes the plates 0, 0, small vessels, etc., which are sub- 
jected to the impact of the water discharged. The rotating force 
is then equal to the difference between the reaction within the 
wheel and the impulse without it. We find, in accordance with 
the theory, that the wheel stands still, when the stream issuing 
from it impinges upon a plane plate at right angles to the direction 
of the water, or when it flows into a vessel filled with water. If the 
stream strikes obliquely against plane-plates or against convex sur- 
faces, the wheel moves in the direction of the reaction, and if it is 
received by a concave surface, the wheel turns in the direction in 
which the water issues from the orifice. 



1020 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 506. 



§ 506. Water-meters. — More recently water-meters (Fr. 
compteurs hydrauliques ; Ger. Wassermesser) have been mucli 
used for measuring running water. They are put in motion by 
the reaction of the water discharged, and consist essentially of a 
reaction wheel or turbine. An ideal representation of the cross- 
section of such a wheel is given in Fig. 857. The water to be 
measured flows through a tube A into the centre of the wheel B B, 

and passes through 4 ca- 
FiG. 857. nals C B, G B . . .to i\iQ 

exterior circumference, 
where it is discharged 
into the case D E, from 
which it is conveyed 
away by a tube E F. The 
shaft }V of this wheel 
carries a pointer Z, or 
rather a train of wheel- 
work, which indicates the 
number of revolutions 
of the wheel, and by it 
the volume of the water, 
which flows through it in any given time ; for this volume is pro- 
portional to the number of revolutions. If h denotes the height of 
a column of water which measures the loss of pressure of the water 
in passing through the wheel, Q the discharge per second, c the ve- 
locity of efflux, and v the velocity of the wheel in the opposite 
direction, we have c^ — v^ = % g li, and the rate of work of the 

wheel 

(c - v) 




L = 



9 



V Cy(see § 505). 



If R is the resistance of the wheel, in consequence of the fric- 
tion on the bearings, etc., we can put L := B v, and from it we 
obtain the formula 



^ = (-^>r, 



or, if F denotes the sum of the areas of all the orifices of efflux, so 

Q 



that Q = F c OT c = 



F 



, we can put 



I — -, from which we obtain 
/ 9 

F Qy 



V 






§506.] THE IMPULSE AND RESISTANCE OP FLUIDS. 



1021 



If R were null, or at least very small, we could put v = ~, ox 

assume the velocity v of rotation to be proportional to the discharge 
Q, which indeed it should be. If, on the contrary, R — ip Vj or ii 
the resistance of the wheel increase with v, we will have 



V + 



i^gv Q 



= -.-. or 



V = 



Q 



^ g^ 



, approximatively 



Q 



' \ Q yr 



V Q yl 
If, then, the resistance R of the wheel is not very small, the 
Telocity of rotation of the wheel is less than when R is null or 
negligible, and the instrument indicates too small a discharge. 

If we put V = 0, we obtain for a discharge Q^ the correspond- 
ing velocity of efflux 

g_R^ 

and we can then put, approximatively at least, 

V ^ c — c^ and 



7T i T in 



0> 



r denoting the radius of the wheel, u the number of its rotations 
and jJb a coefficient to be determined by experiment. 

Within the last few years Siemens's water-meter has come into 
very general use; its principal parts are represented in cross- 
Bection in Fig. 858. The water which enters from A passes 

Fig. 85a 




1022 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 506. 



Fig. 859. 



through the pipe B B into the wheel C C and is carried by the 
revolving tube D D into the case E E, from which it is carried off 
by the pipe F. The shaft W of the wheel passes upwards through 
a stuffing-box and sets a train of wheel-work in motion by means 
of an endless screw fastened to its end. The wings Ic, h upon the 
iyheel assist in regulating its motion of rotation by the resistance 
>vhich they experience in movmg in the water. 

The reaction wheel can be constructed in such a manner that- 
Bvery time it makes a revolution it will allow a certain quantity of 
water to pass through. To accomplish this object, the wheel B A B, 

Fig. 859, is partially immersed in water, 
so that, when turning, the spiral tubes 
are alternately filled with air and water. 
Here also the water is conducted by a 
pipe into the centre of the wheel, and 
from thence by spiral pipes into the 
free space of the case E F, from which 
it flows away through the pipe F. The 
surface of the water in the interior of 
the wheel is at a distance li above that 
of the water in the case ; hence, if the 
wheel turns in the direction indicated 
by the arrow, as soon as the orifice D 
arrives at the level of the water in the 
interior, the water begins to discharge, and in so doing reacts with, 
a certain force P, by which the rotation of the wheel is main- 
tained. If y is the volume of the water contained in one of the 
spiral pipes, and n the number of these canals, the discharge per 
second, when the number of rotations per minute of the volume of 

, - , . . ^ nuY 

the water is -x^, is ^ — — -p^^r-, 

oO 

Remark. — An account of Siemens' water-meter is given in the " Zeit- 

schrift des Vereines deutscher Ingenieure," Vol. I, 1857, in which Jopling's 

water-meter (in which the water is gauged) is also described. See also the 

paper : " Siemens and Adamson's Patent Water Meter." A very peculiarly 

constructed water-meter of the nature of a reaction wheel is described in 

the "Genie industrielle," Tome XXI, No. 126, 1861, under the name: 

*' Compteur hydraulique pour la mesure d'ecoulement des liquides par 

Guyet." Two water-meters are described in the English work " Hydrau- 

lia," by W. Matthews. A compteur hydraulique used at the railroad station 

at Chartres is described in the " Bulletin de la Societe d'encouragement,'* 

51 year (1852) Uhler's apparatus for measuring fluids is treated of in 




§ 507.] 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1023 



Dingler's Journal, Vol. 161. A description of an apparatus for measuring 
the quantity of spirit made in distilleries is contained in the " Mittheilnn- 
sren des Gewerbevereines for Hannover," new series, 1861. 

For a description of several kinds of water-meters, see "The Transac- 
tions of the Institution of Mechanical Engineers," 1856 (Tr.). 

§ 507. G-as-meters. — The so-called wet gas-meters (Fr. cornp- 
teiirs a gaz ; Ger. Gasmesser or Gasuliren) are, like certain water- 
meters, small wheels with spiral canals, which are more than one- 
half immersed in water and are put in motion by the reaction of 
the gas i^assing through them ; each spiral canal transfers a certain 
volume of gas from the inside to the outside. The essential parts 
of such a gas-meter are shown in the two sections of Fig. 8G0. 

The gas, which arrives, 
Fig. 860. enters by a bent pipe A 

^ " into the interior of the 

measuring wheel B B, 
in which it depresses 
the surface of the Avater 
~ a certain distance li, 
which depends upon 
the tension of the gas. 
passing through the in- 
strument. From this 
central chamber it enters successively the spiral canals, fills them 
almost entirely and, finally, passes out through the orifices at the 
circumference into the case G G, from which it is conducted by a 
pipe H to the point, where it is to be used. As we wish every 
spiral canal of the measuring wheel to carry over a certain definite 
quantity of gas at each revolution, we must so arrange the appa- 
ratus that at least one of the orifices of a canal shall always be 
under water ; for in that case, when the gas is filling the canal, 
there is no efflux, and during the efflux no gas can enter it. The 
volume of gas F, passed by one spiral canal, is consequently a defi- 
nite one, and we can, therefore, put the discharge per minute 

when the wheel makes n revolutions per minute. If we denote the 
height of the barometer in the gas leaving the machine by h, that 
in the gas entering it is J + h, and, therefore, according to Ma- 
riotte's law, the quantity of air in one spiral canal, measured at the 
pressure of the gas after it has left the measuring wheel, is 




1024 GENERAL PRINCIPLES OF MECHANICS. [§507. 

'h + h 



V, 



m -' 



consequently the quantity of gas, which passes from the wheel into 
the exterior case when the outlet of one of the spiral canals risio 
from the water, is 



When this quantity streams into the case the mechanical effect 

set free is . ^t- -, 1^ + ^^^ 

A 



V, I i^-±-') 



(see § 388), and since ^ is small, we can put 



^m=^{^-t)=p 



hence, if the heaviness of the suhstance, with which the manometer 
is filled, is y, we have p = (b -{- h) y = by, and therefore A —Vliy. 
One portion of this mechanical effect is expended in turning 
the wheel, and the rest in producing an eddy. The first portion 
is determined by the expression 

_ {c-v)v h 
A,- g 'b^^'[ 
in which h denotes the mean height of the manometer, c the mean 
velocity of efflux, v the velocity of the wheel at its circumference 
and yi the heaviness of the gas discharged. If R is the resistance 
of the wheel, reduced to its circumference, and r its radius, we 
have the required mechanical effect 

Ai = R, , and therefore we can put 

n ^ 

(C — V) V h ^^ ^TTT ^ . _ 60 V 

^^ — . T y y\ = Ri or since 2 77- r = , 

g n u 



c - V h ^^ 60 R 



g h n u 

hence it follows that the velocity of rotation, corresponding to the 
distance A between the two surfaces of water, is 

gh 60 R 



V =^ c — ~ 



hVyi' n u 

and that the number of revolutions of the meter per minute is 

30 / 60 ql)R\ 

TT r V n u V hyxf 

ApproximatiYely we have c = y 2 ^ , when y denotes the 



§508.] 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1025 



heaviness of the substance with which the manometer is filled. 
The volume of gas passing per minute is 

and it is proportional to the number of revolutions u. 

§ 508. Ne"wer G-as-meters. — Instead of placing the spiral 
canals of a gas-meter in a plane perpendicular to the axis, we can 
wind them round it like the thread of a screw. The action of 
such a gas-meter is shown by the two sections I and II, Fig. 861, in 
which D D represents the surface of the water at the front and E E 

Fig. 861. 




that at the back of the measuring wheel, which is a horizontal 
drum. The orifice A of the spiral canal A B opens into the 
chamber, which is in front of the drum, and receives the gas, which 
is arriving ; the orifice B, on the contrary, delivers the gas into 
the chamber at the back of the drum, from which it is carried off 
by a pipe. In JFig. 861, *I, the different positions of a spiral canal, 
viewed from in front of the wheel, are represented. Fig. 861, II, 
on the contrary, represents the various positions of the canal as 
seen from the rear of the wheel. In consequence of the rotation 
of jthe wheel, in the direction indicated by the arrow, around the 
honzontal axis (7, the inlet orifice A in (1, 1) is just emerging from 
the water in front, while the outlet B is just entering the water in 
the rear, in (I, 2) and (I, 3) the arcs A 0, A of gas have entered 
through the orifice A, and in (I, 4) the orifice has re-entered the 
water, so that after a certain quantity Y has been received into the 
canal, the entry of the gas is cut off. Shortly afterwards the orifice 



102G 



GENERAL PRINCIPLES OF P»IECHANICS. 



[§ 508. 



B rises, as is represented in (II, 1), from the water in the rear of 
the drum and the discharge of the gas, which had previously been 
taken in, begins, and it is in full operation in tlie positions (II, 2) 
and (II, 3). When a new revolution begins, B re-enters the 
water in the rear of the drum, as is represented in (II, 4), and the 
gas again begins to fill the canal. During half a revolution of the 
spiral canal A B, an arc of gas A (I, 4), which is at the greater 
tension h + li, enters the former and during the second half of 
the same it is transferred to the space beyond the wheel, where the 
pressure is less. In passing from the greater pressure to the less, 
the mechanical effect A —Vhyi^ set free ; a portion of this is 
expended in moving the wheel, as was shown in the foregoing- 
paragraph. The general arrangement and action of such a gas- 
meter can be better understood from the ideal representation in 
Fig. 862. The gas is first introduced by means of a bent tube A 
into a chamber B B, which communicates in the middle around 
the axis of rotation C with the water in the case E F G, but upon 
the exterior circumference, where the spiral tubes enter it, it is air- 
tight. The drawing shows the spiral canal H K to be receiving 
gas from B B and the canal L M, which a short time before had 
received a certain volume of gas, to be discharging it at M into the 
upper space in the case E F G, from which it is carried away by 
the pipe F. By this arrangement of the meter the gas in the first 
chamber is cut off entirely by the water from that in the rear 
chamber, and, therefore, the packing, which causes great loss of 
force, is rendered unnecessary. The other end D of the axis D 
of the wheel has a couple of turns of a screw cut upon it, by means 
of which the train of wheels of the counting apparatus is set in 

motion. 

Fig. 862. Fig. 863. 





§ 508.] 



THE IMPULSE AND RESiSrAlNCE OF FLUIDS. 



1027 



Fig. 864. 



Crossley's gas-meters, which have come into very general use, 
are constructed according to the principles explained above ; but 
their spiral canals are not tube-shaped, but real chambers or cells 
with spiral partitions and with triangular inlet and outlet orifices, 
which are made by bending out the end surfaces. Fig. 863 is a 
perspective view of such a wheel with the cover removed ; it con- 
sists of 4 pieces of sheet iron like that represented in Fig. 864. 
Ai, ^1<, A-i, A^ are the inlet orifices, Bi, Bo . . . the outlet orifices 
and Ci, 0-2, C-^ . . . the partitions of the measuring wheel which 
turns around the axis TJ D. Fig. 865 is an elevation of the gas- 
meter with the exterior drum or case ; we observe at K the bent 
tube, which conducts the gas into the chamber, and at Z the pipe, 

which carries oS the gas 
from the upper space A A 
of the case of the meter. 
The gas does not flow di- 
rectly into K, but the pipe 
Scarries it first into a cham- 
ber F, from which it passes 
through the conical valve i 
into the chamber G, where 
it enters the upper part of 
the vertical pipe R, through 
which it is conducted into 
the bent tube A". The sur- 
face of the water in the 
chamber G reaches exactly 
to the top of the pipe H, 
through which the super- 
fluous water overflows into 
a reservoir L. In order, on 
the other hand, to prevent 
the water from sinking too 
low, a float is jTlaced in the 
chamber, which, when it 
sinks, carries the valve i with 
it and closes the opening, 
when the float has sunk a 
certain distance. The dis- 
charge of gas then ceases en- 
tirelv, and we are thus noti- 




FiG. 865. 




1028 



GENERAL PRINCIPLES OF MECHANICS. 



[§ 508. 



fied that it is necessary to fill the meter with water through an 
orifice M, that opens into a chamber N, which coinmunicates, at 
the bottom only, with the water space. 

'Fig. 866 is transverse elevation of the front of such a meter, in 
which are to be seen not only the chamber N with the orifice M, 
but also the clockwork of the counting apparatus, which is set in 
motion by an endless screw upon the axle of the drum and a ver- 
tical shaft with a cog-wheel upon it. 

An important resistance to the motion of Crosley's gas-meter is 
that occasioned by the entry and exit of the water through the 
narrow triangular orifices. We can calculate from the area F of 
an inlet or outlet orifice and from the discharge per second, which 
can be put equal to the volume Q of the gas, the velocity of exit 

Fig. 866. 




and entrance v^ = -^, and consequently the corresponding loss of 
mcQhanical effect per second 

Remark. — Particulars upon the subject of gas-meters can be found in 
Schilling's " Handbuch der Steinkohlengasbeleuchtung," and Heeren's 
article " die Einrichtung der Grasuhren " in the " Mittheilungen des Ge- 
werbevereins fur das K. Hannover," year 1859. A new gas-meter by Han- 
sen is described in the " Journal der Gasbeleuchtung," 1861. 



§ 509.] 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1029 



§ 509. Action of Unlimited Fluids. — If a body has a mo- 
tion of translation in an unlimited jiuid, or if a body is placed in a 
moving fluid J it is subjected to a pressure, which is dependent upon 
the form and size of the body as well as upon the density of the 
fluid and the velocity of one or other of the masses ; in the former 
case it is called the resistance and in the latter the impulse of the 
fluid. This hydraulic pressure is principally due to the inertia of 
the water, whose condition of motion is changed when it comes 
into contact with a rigid body, and also to the force of cohesion of 
the molecules of water, which are partially separated from and 
moved upon each other. 

If a body A (7, Fig. 867, is moved in still water, it pushes a 
certain quantity of water, the pressure of which is increased, before 
it. As the body progresses the quantity of water on one side is 
increased, while upon the other it is constantly flowing away, and 
the particles lying immediately contiguous to the surface A B 

Fig. 867. Fig. 868. 





assume a motion in the direction of this surface. If a stream of 
ivater encounters an obstacle A C, Fig. 868, which is at rest, the 
pressure of the water in front of it is increased, the molecules of 
water are diverted from their original direction and move along 
the front surface A B. When the particles of water have reached 
the edges of the front surface, they turn and follow the sides of 
the body, until they arrive at the back surface, where they do not 
immediately reunite, but assume first an eddying motion. We see 
that the general relations of the motion of the molecules, which 
surround the body, are the same for the impulse of water as for the 
resistance to a body moving in the water ; but there is a difference 
in the eddies, when the body is short ; for in the latter case the 
eddies occupy less space than in the former. The velocity of the 
molecules of water increases gradually from the centre of the front 
surface to the edges, where a contraction generally takes place and 
where the velocity is a maximum, it decreases as the water passes 
along the sides and becomes a minimum when the water arrives 
at the back surface and begins its eddying motion. 



1030 GENERAL PRINCIPLES OF MECHANICS. [§510. 

§ 510. Theory of Impulse and Resistance. — The normal 
pressure of still or moving water upon a body moved or immersed 
in it is very different at different points of the body. This pres- 
sure is a maximum at the centre of the front surface and a mini- 
mum ill the centre of the rear surface and at the beginning of the 
sides ; for at the first point the water flows towards the body, and 
at the latter points it flows away from it. If the body is, as we 
will suppose in what follows, symmetrical in reference to the 
direction of motion, the pressures at right angles to this direction 
balance each other, and we must, therefore, consider only the 
pressures in the direction of the motion. But since the pressure 
upon the rear surface acts in an opposite direction to those upon 
the front surface, it follows that tlie resulting impulse or resistance 
of the tvater is equal to the difference between the pressures upon the 
front and iach surfaces. 

Although we cannot determine a priori the intensity of this 
pressure, yet, as the circumstances are very similar to those of the 
impact of an isolated stream, we can at least assume that the gen- 
eral law of the impact of an unlimited stream does not differ very 
much from that of an isolated stream. If F is the area of surface 
which an unbounded stream, whose heaviness is y and whose velo- 
city is ?', encounters, we can put the corresponding impulse or hy- 

in which ^ denotes an empirical number dependent upon the shape 
of the surface. This formula can be applied not only to the front, 
but also to the rear surface. But in the latter case, where the 
water tends to separate itself from the body, the expression becomes 
negative. Now \i F h y is the hydrostatic pressure (§ 690) against 
the front and against the back surfaces of a body, the total pressure 
against the front surface is 

P, = Fhy\-^,.^Fy, 

and that against the back surface is 

hence the resulting impulse or resistance of the water is 

when we put (^^ -\- ^^ = ^. 

This general formula for the impulse and resistance of an un- 
limited stream is also applicable to the impulse of wind and to the 



§511.] THE IMPULSE AND RESISTANCE OF FLUIDS 1031 

resistance of the air. Here, however, besides the differeis ^ of the 
aerodynamic pressures upon the front and rear surfaces, a difference 
in the aerostatic pressure also exists, which is due to the fact that 
the air at the front surface has a greater heaviness (y), in conse- 
quence of its greater tension, than that at the rear surface. For 
this reason, at least when the velocities are great, as in the case of 
musket and cannon balls, the coefficient of resistance of the air is 
greater than that of water. 

Remark. — A peculiar phenomenon attends the impulse and resistance 
of an unUmited medium (water or air), viz., a certain quantity of water or 
air attaches itself to the body, the influence of which is shown by the vari- 
able motion of the body, which, e.g., is very evident in the oscillations of 
a pendulum, The quantity of air or water which attaches itself to a sphere 
is 0,6 the volume of the sphere. For a prismatic body, moving in the di- 
rection of its axis, the ratio of these volumes is 

= 0,13 + 0,705 -j~, 

in which I denotes the length and F the cross-section of the body. This 
ratio, which was first determined by du Buat, has been fully confirmed by 
the later experiments of Bessel, Sabine and Bailly. 

§ 511. Impulse and Resistance against Surfaces. — The 

coefficient ^ of resistance, or the number by which the height ^— 

due to the velocity must be multiplied, in order to obtain the height 
of the column of water which measures the hydraulic pressure, is 
very different for bodies of different form ; it is determined approx- 
imatively only for plates, which are placed at right angles to the 
stream. According to du Buat's experiments and those of Thi- 
bault, we can put for the impulse of water and air aganist a plane 
surface at rest ^ = 1,86, while, on the contrary, we can assume with 
less certainty for the resistance of the air and water to a plane sur- 
face in motion ^ = 1,25. In both cases about two-thirds of the 
action is upon the front and about one-third upon the rear surface. 
The values, found for the resistance offered by the air to a body re- 
volving in a circle by Borda, Hutton, and Thibault, vary much 
from each other. The latter found with a rotating plane surface, 
the area of which was 0,1 square meter, the resistance 
P = 0,108 Fv% whence 

f = 0,108 . ^ = 0,108 . ^^ = 1,70. 

y 1,25 

This coefficient is, according to these experiments, almost con- 



1032 GENERAL PRINCIPLES OF MECHANICS. [§ 511. 

stant, when the angle a formed by the surface with the direction 
of the motion is not less than 45°. When the angle is less than 
45°, the coefficient diminishes with this angle of impact, and for 
a = 10°, ^ is only = 0,53. According to the researches of Didion, 
etc., we have for the resistance of rotating plane surfaces, whose 
areas are 0,2 . 0,2 = 0,04 square meters, 

^ = (0,1002 + 0,0434 v~') . ?^ = 1,573 + 0,681 v-% 

in which t^ must be given in meters. 

For a plane surface, whose area was one square meter, Didion 
found, when the motion was vertical, the coefficient of resistance 

^ = (0,084 + 0,036 V-') . ^ = 1,318 + 0,565 v-% 

while Thibault, on the contrary, found for such surfaces, when 
their area was 0,1 to 0,2 square meters, 

^ = (0,1188 + 0,036 V-') . ^ = 1,865 + 0,565 v-\ 

The foregoing formulas hold good only when the motion of the 
surface is uniform ; if the motion is variable, they require an ad- 
dition. If the velocity of a body which is moving in a resisting 
medium changes, the quantity of the fluid moved by the body or 
carried along with it varies ; the resistance is, therefore, dependent 
upon the acceleration p. According to the experiments of Didion, 
etc., with a surface whose area was 1 square meter, and with one 
whose area was } square meter, which were moved in a vertical line, 
the resistance was 

P := (0,084: v' + 0,036 + 0,164 j(?) F; hence 

^ = [0,084 + (0,036 + 0,1U p)v-'] . ^-^^ 

= 1,318 + (0,565 + 2,574)y-^ 
We must also remember that for variable motion the mean 
square of the velocity is different from the square of the mean 
velocity. 

The impulse and resistance of an unhmited medium is increased 
when the surfaces are hollowed out or provided with borders ; but 
we have as yet no general data concerning the subject. 

For a parachute, whose cross-section was 1,2 square meters and 
whose mean diameter was 1,27 meters and whose depth was 0,430 
meter, Didion, etc., found for an accelerated motion, during which 
the hollow surface was in front, 

P == (0,163 v' + 0,070 + 0,142 p) F, whence 
^ = 2,559 + (1,099 + 2,229 p) v-\ 



§512.1 THE IMPULSE AND RESISTANCE OF FLUIDS. 1033 

§ 512. Impulse and Resistance against Bodies. — The im- 
pulse and resistance of water against prismatical bodies, whose axis 
coincides with the direction of motion, decrease when the lengths 
of the bodies increase. According to the experiments of du Buat 
and Duchemin, the impulse upon the front surface is constant, and 
the action upon the rear surface alone is yariable. The coefficient 
^1 = 1,186 corresponds to the former; but when the relative 

lengths are ^ _ n i o q 

— -0, 1, ^, 3, 

the total action is 

^= 1,86; 1,47; 1,35; 1,33. 

If the ratio between the length and the mean width V^ be- 
comes greater, the coefficient ^ again increases in consequence of 
the friction of the water upon the sides of the body. The reverse 
is true of the resistance of the water. In this case, according to 
du Buat, the constant action against the front surface is ^j = 1, and 
the total action for 

-^-0, 1, 2, 3, is 

^ = 1,25; 1,28; 1,31; 1,33; 
so that for a prism three times as long as Avide the impulse of the 
water is the same as the resistance. 

The experiments of Newton, Borda, Hutton, Vince, Desaguil- 
liers and others with round and angular bodies leave much uncer- 
tain and undetermined. It appears that for moderate velocities the 
coefficient of resistance of spheres can be put = 0,5 to 0,6. But 
when the velocities are greater and the motion takes place in the 
air, we can put, according to Robins and Hutton, for the velocities 
V = 1, 5, 25, 100, 200, 300, 400, 500, 600 meters, 
^ = 0,59 ; 0,63 ; 0,67 ; 0,71 ; 0,77 ; 0,88 ; 0,99 ; 1,04 ; 1,01. 

Duchemin and Piobert have given particular formulas for the 

increase of this coefficient of resistance. According to Piobert the 

resistance to a musket ball in the air is 

P = 0,029 (1 + 0,0023 v) F v" kilograms, whence 

^ = 0,451 (1 + 0,0023 V), 
For the impulse of water against a ball, Eytelwein found 

C = 0,7886, 
while, on the contrary, according to the experiments of Piobert, 
etc., made with cannon balls 0,10 to 0,22 meters in diameter, the 
resistance to the balls in water is 

P — 23,8 F v" kilograms ; hence we can put 

<r = 0,467. 



1034 



GENERAL PRINCIPLES OP MECHANICS. 



[§512. 



The coefficients of resistance for bodies partially immersed are 
different from those for bodies entirely surrounded by water. For 
2i, floating prismatic body five to six times as long as wide and mov- 
ing in the direction of the axis, ^ should be put equal to 1,10. If 
the body is sharpened in front by two vertical planes like ABC, 
'Fig. 869, s increases with the angle A C A — (3, and we have 



fori3 = 


180° 


156° 


132° 


108° 


84° 


60° 


36° 


12° 


^ = 


1,10 


1,06 


0,93 


0,84 


0,59 


0,48 


0,45 


0,44 



If, on the contrary, the rear portion A O By Fig. 870, is sharp- 
ened, and if the angle BCB — (3,we have 

Fm. 869, Fig. 870. 







for^ = 


180° 


138° 


9G° 


48° 


24° 


C- 


1,10 


1,03 


0,98 


0,95 


0,92 



When both front and rear portions of the floating body are 
sharpened, ^ becomes still smaller. For river steamboats, ^ = 0,12 
to 0,20 and for large ocean steamers, ^ =: 0,05 to 0,10. 

Eemakk. — This subject is treated at length by Poncelet in his "Intro- 
duction" cited above, and by Duchemin and Thibault in their " Recherches 
experimentales, etc." The subject of the resistance to floating bodies, par- 
ticularly ships, and also that of the impulse of the wind against wheels, 
will be treated in the second and third volumes. 

Example. — If, according to Borda, we put the resistance and impulse 
at right angles to the axis of a cylinder |- that against a parallelopipedon, 
which has the same dimensions as it, we have the coefiicient of resistance 

C = 1 . 1,28 = 0,64 
and the impulse against the same 

= I- . 1,47 = 0,735. 
If we apply these values to the human body, the area of the cross- 
section of which is 7 square feet, we find the resistance and impulse of the 
air against it 

P = 0,64 . 0,0155 . 7 . 0,086 v^ = 0,00597 v^ 
and 

P = 0,735 . 0,0155 . 7 . 0,086 v'' = 0,00686 v\ 



§513.] THE IMPULSE AND RESISTANCE OF FLUIDS. 1035 

For a velocity of 5 feet, the resistance of the air is, therefore, only 
0,00597 . 25 = 0,1493 pounds, and the corresponding work done per 
second is = 5 . 0,1493 = 0,746 foot-pounds ; for a velocity of 10 feet, the 
resistance is 4 times and the expenditure of mechanical effect 8 times as 
great, and for a velocity of 15 feet the resistance is 9 times and the work done 
27 times greater. If a man moves with the velocity 5 feet against a wind, 
whose velocity is 50 feet, he has to overcome a resistance 0,00686 . 55^ = 
20,75 pounds, which corresponds to a velocity of 50 + 5 = 55 feet, and 
to perform an amount of work equal to 20,75 . 5 = 103,75 foot-pounds. 

§ 513. Motion in Resisting Media. — The laws of the mo- 
tion of a tody in a resisting medium are not very simple ; for the 
force in this case is variable, increasing with the square of the 
velocity. From the force P, which is drawing the body onwards, 

and from the resistance Pi = (^ . -— F y, offered by the medium, 
we obtain the motive force 

but since the mass of the body is Jf = — , its acceleration is 

or if we denote ^f^ by -^, or put y ~tt- = w. we have 

1 g P ^ w ^ ^ ^ Fy ' 

The maximum velocity which the body can assume is 



If the motive force P is constant, the motion approaches grad- 
ually a uniform one; for the acceleration becomes smaller and 
smaller as v increases. 

Now the velocity v increases, when the acceleration is j9, in an 
element of time t a quantity k — p r, hence we can put 

'^ = \}- 1^)'] -Qff''^^^ inversely 

G K 



9 



[' - Q] 



1036 GENERAL PRINCIPLES OF MECHANICS. [§513. 

In order to find the time, corresponding to a given variation of 
velocity, let us divide the difference v^ — v^ between the initial and 
the final velocities in n parts and put such a part 

n 
and then calculate from it the velocities 

Vi = v^ + K, v^ = v^ + 2 K, Vi — v^ -\- Z K, etc., 
substituting these values in Simpson's formula, we obtain the 
required time, when we assume four divisions. 



+ 






The space described in an element r of time (§ 19) is 
a = V T, or smce we can put r = -^ 

V K 

(7 = — , or 

V K G 

a = 



-(-) 



Pg 



By employing Simpson's rule we find the space described, while 
the velocity changes from v^ to v^, to be 

. 6^ V^ - t\ I V, AVr 



+ 



"'I' -(5) ■-©■ 
'-(5)' '-(5)' -©■)■ 



The calculation is of course more accurate when we make 6, 8, 
or more divisions. This formula allows us to take into account 
the variability of the coefiicient of resistance, which is necessary for 
high velocities. Por the free fall of a body in air or water P = G, 
the apparent weight of the body, and for motion in a horizontal 
plane P = 0, or more correctly, equal to the friction / G. Since 
this is a resistance, it must be introduced as a negative quantity in 
the calculation ; hence we must put 



§513.J THE IMPULSE AND RESISTANCE OP FLUIDS. 1037 

Po = - (P + Pi) and 

Since in this case there can be no question of an increase, but 
only of a decrease of velocity, we must substitute v^ — v^ in the 
above formulas instead of v^ — v^. 

When a body is impelled by a force, such as its own weight, the 
motion approaches more and more to a uniform one, and after a 
certain time it may be considered as such, although it never will be 



v' 



really so. The acceleration p becomes = 0, when ^ —- Fy = P„ 
or when 



A body falling freely in air approaches more and more to this 
result without ever attaining it. 

Example. — Piobert, Morin and Didion found for a parachute whose 
depth was 0,31 times the diameter of the opening, the coefficient of resist- 
ance C = 1,94 . 1,37 = 2,66, From what height can a man weighing 150 
pounds descend with such a parachute weighing 10 pounds and with a 
cross-section of 60 feet, without assuming a greater velocity than that he 
attains when he jumps down 10 feet? The latter velocity is ?) = 8,025 V 10 
= 25,377 feet, the force P == G^ = 150 + 10 = 160 lbs., the surface i^ = 60 
feet, the heaviness y = 0,0807 pounds and the coefficient of resistance 

C = 3,66, hence 

1 2,66 . 60 . 0,0807 1,33 . 3 . 0,0807 



w"" 64,4 . 160 64,4 .4 ~ ' 

and -Y = 0,00125 . 25,377' = 0,805 



= 0,00125 



If we assume six divisions, we obtain 
1 - — , = 0,977639 ; 0,91055 ; 0,79875 ; 0,64222 ; 0,44097 ; 0,195, 

and ^ = 0; 4,326; 9,290; 15,886; 26,343; 47,958, and 130,138; 

1- — 

hence, according to Simpson's rule, we have the mean value 

= (1.0+4. 4,326 + 2 . 9,290 + 4 . 15,886 + 2 . 26,343 

474 084 
+ 4 . 47,958 + 1 . 130,138) : (3 . 6) = — -^ — = 26,338 ; 

lb 

and the required space, through which he can fall, is 
g _. ^iJZJi times the mean of — ^ = ^^'^JJ~ ^ . 26,338 = 20,76 ft 



1038 



GENERAL PRINCIPLES OF MECHANICS. 



[§514. 



Tbe corresponding duration of the fall, since the mean value of 



1 - 



1? 



IS 



= (1.0+4. 1,023 + 3 . 1,098 + 4 : 1,252 

+ 2 . 1,557 + 4 . 2,268 + 1 . 5,128j : 18 = 1,589, is 



t = -~^r ' 1'^^^ = ^'^^ seconds. 



Remark. — If the coefficient of resistance is constant, we obtain by the 
aid of the Calculus for the case of a body falling freely 



G 



and 



in which 



fit + 1} \ei^t + 1} ^ "^ ' ^Fy 



~~ \w- — -^7 * 2 ^ ' 



and e denotes the base of the Naperian system of logarithms and I the Na- 
perian logarithm. 

§ 514. Projectiles. — We have already studied the motion of 

projectiles in vacuo and found in § 39 the path or trajectory to be 

Fig. 871. ^ parabola. We can now investigate this 

motion in a resisting medium, e.g., the 

motion of a body projected in the air. 

The path of a body projected through 
air is certainly not a parabola, as is the case 
when it is projected in vacuo, but an un- 
symmetrical curve ; the portion of the tra- 
jectory, where the body is rising, is not so 
steep as that where it is falling, as can be 
seen from what follows. During the instant r the body, loliich is 
rising with a velocity v in the direction A T, Fig. 871, describes, 
in consequence of its inertia, the space 

A ^= s = V T, 
and, in consequence of gravity, the vertical space 

and the first space is diminished by the resistance ^ — F y ot 
the air an amount, which can be determined by the expression 




§ 514.] 



THE IMPULSE AND RESISTANCE OF FLUIDS. 



1039 



If we put s 



Fy 



%G 



^— Fy 

G '2 ^ %G' % 
— II, we have more simply 

Q = ii 



V T 



2 



The fourth corner i? of the parallelogram F Q R, constructed 
with F and Q, gives the position which the body occupies at 
the end of the time r, while F is the place which the body would 
have occupied at that moment, if the air offered no resistance. 
The path A R oi the projectile passes, therefore, below the para- 
bola, which the body would have described in vacuo. 

In like manner we have for a body descending with the initial 
velocity v in the direction A T, Fig. 872, the spaces described si- 
multaneously in the time r 



A 



V r, 

^2 



P == ^ y, and 



Q = fiv' 



r' 

y 



and from the above we obtain again the position R occupied by the 
body at the end of this time, and the position F which it would 
have occupied, if its motion had taken place in vacuo. The path A R 
described in this case passes also below the parabolic path A F, 
which the body would have followed, if the air opposed no resistance. 
If the angle of incHnation, at which a body rises with the initial 

Fig. 873. 





Telocity v from A, is T A X = a, Fig. 873, the initial co-ordinates 
or velocities in the direction of the axes are 



1040 GENERAL PRINCIPLES OF MECHANICS. [§514. 

u =^ V COS. a and 

w = V sin. a, 
and we have for the position E of the moving body, after an instant 
r, the abscissa 

A M = X = A Q COS. a = Iv r — ) cos. a 

= (1 — I V r cos. a, 

and the ordinate 

M R = y = A Q sin. a — Q B = (l — ^^^) v r sin. a — ^~-, 

The velocity in the direction of the abscissa is 

B Ui = Ui=^ V COS. a — fjb v^ T COS. a = (1 ~ ^ v r) v cos. a, 
and that in the direction of the ordinate is 



B Wi = Wi = V sin. a — pbv' T sin. a — g r ={1 — ii v r) v sin. a—gr. 
From the two velocities we obtain the angle of inclination 
TiB Xi = ai of the path at B by means of the formula 

tang, a^ = — = tang, a — ^^-— , 

Ui (1 — fj^v r) v COS. a 

and the velocity in the direction of the curve is 



Rvi=Vx— Vux + Wi = V{l—^vryv^—2 (1 — fivr) V g r sin. a+g^ t''. 
By repeated application of this formula, we can find the course 
of the whole trajectory of the projectile. If, e.g., we substitute in 
the above formulas for x and y, instead of a and v the values for a^ 
and Vi obtained from the last equation, we obtain the co-ordinates 
Xi and yi of a new point referred to B, etc. 

Example. — A massive cast-iron cannon-ball, whose diameter is 2 r = 4 
inches, is projected at an angle of elevation a = 25° with a velocity « = 
1000 feet ; required the position of the same after ■^-^, -^^, -j?- of a second, etc. 

Since the weight of a cubic foot of air is 0,080728 pounds and that of a 

cubic foot of cast iron is 444 pounds, we have 

Fy ■kt'' y y 0,080728 

fz = -^.f=- JL^ ^= 3 JL ^^ 1 , 6.— 4^:^ — C= 0,000409094 C, 

and, therefore, for v = 1000 feet, for which C = 0,9 (see § 512), we have 

fi = 0,0003682. 
If we take r = 0,1 seconds, we obtain 
aj = (1 — 0,0003682 . 1000 . 0,05) 100 cos. 25° = 0,98159 . 90,63 = 88,96 feet, 

y = 0,98159 . 100 sin. 25° - 33,3 . -~ =0,98159 . 42,26 - 0,16=41,33 feet, 
and 



§514.] THE IMPULSE AND RESISTANCE OF FLUIDS 1041 

33,2.0.1 ^..... 3,32 

Un^. a^ = tang. 25° - -. Trrk^^KT-iTf^^ = 0,46631 - 



(1 — 0,03682) . 906,3 ' 0,96318 . 906,3 

= 0,46631 — 0,00369 = 0,46262 ; 

hence the angle of elevation is 

a = 24° 50', 

and the velocity in the curve is 

v^ = V(0,96318 . 1000)'^ - 2 . 0,96318 . 1000 . 32,2 . 0,04226 + (3,22)» 

= V927716 — 2621 + 10 = 7935105 = 961,82 feet. 
If we again take r = 0,1 second, we have, since for v = 962 feet, ^ = 
0,88, and consequently ft = 0,88 . 0,000409094 = 0,00036, 
x^ = (l — 0,00036 . 961,8 . 0,05) . 96,18 cos. 24° 50' 

= 0,9827 . 96,18 . 0,9075 =-. 85,77 feet, 
y^ = 0,9827 . 96,18 sin. 24° 50' - 0,161 = 39,53 feet, 

and 

3 22 

tang, a, = ta.ng. 24° 50' - o,^653TT96i;Fc... 24° 50' 

= 0,46277 - 0,00382 = 0,45895, 
whence 

c = 24° 39' and 

« = V(0,96537 . 961,8)'^ — 2 . 0,96537 . 961,8 . 32,2 . 0,04300 + (3,32)» 

= V862099 - 2511 + 10 = V859598 = 927,14 feet. 
Assuming once more r == 0,1 and v = 927 feet, we have C = 0,87 

fi = 0,87 . 0,000409094 = 0,0003559, 
and therefore 

ajg = (1-0,0003559 . 927 , 14 . 0,05) . 92,71 cos. 34° 39' =0,9835 . 93,71 . 0,9089 

= 83,87 feet and 

^3 = 0,9835 . 93,71 sin. 34° 39' - 0,156 = 37,87 feet. 

The position of the projectile in reference to the point of beginning is 
determined after 0,3 seconds by the co-ordinates 
X + x^ + x^ = 88,96 + 85,77 + 83,87 = 357,60 feet and 
y +y^ + y^ = 41,33 + 39,53 + 37,87 = 118,73 feet. 

If the air offered no resistance and gravity did not act, we would have 
x + x^ +X2 =ct cos. a = 1000 . 0,3.cos.25°=: 300. 0,9063 = 271, 89 feet and 
y + yt + y2=ctsin.a = 300 . sin. 35° = 300 . 0,4336 = 136,78 feet. 
If we neglect the resistance of the air only, we have 

X + Xj^ + X2 = 371,89 feet and 

o f^ 09 

y + yi+y2 = 126,78 - ^ = 126,78 - 32,3 . -^ = 136,78 — 1,449 

= 125,33 feet. 
66 



APPENDIX. 



THE THEORY OF OSCILLATION. 



(§ 1.) Theory of Oscillation. — A body has an oscillatory or 
vibratory motion (Fr. mouvement oscillatoire ; Ger. schwingende 
Bewegung) or is in oscillation or vibration (Fr. oscillation; Ger. 
Scliwingung), when it describes repeatedly the same path backwards 
and forwards in equal times. We meet with many examples of 
oscillatory motion in nature besides that of the pendulum. The 
most general cause of such a motion is a force which attracts or 
impels the oscillating body towards a certain point. Thus, e.g., 
gravity sets the pendulum in oscillation. If a body, previously at 
rest, can yield without impediment to the action of the force, which 
impels it towards a certain point, the oscillation takes place in a 
straight line ; otherwise it will oscillate in a curve, as a pendulum 
does, where the action of gravity is continually interfered with, 
the body being united to a fixed point. In like manner, if the 
direction of the initial velocity of the body is different from that 
of the motive force, the oscillations will also take place in curved 
lines. 

The simplest and most common case is that where tlie force is 
"proportional to the distance of the body from a certain point C. Let 

C, Fig. 87,4, be the seat of the force, i.e. 
the position of the body when the force 
is =: ; let ^ be the point where the 
motion begins, and let M be the variable 
position of the body. If we denote the 
distance C Mhj x, and by // a constant, 
determined by experiment, we have the 
acceleration of the body at M 
p = fix. 



Fm. 874. 
D 




§2.] THE THEORY OF OSCILLATION. 1043 

and since x decreases an amount d x, when the space A M is in- 
creased by the same quantity, we have for the velocity v of the 
body (see § 20, III) 

^v"^ = — 1 2^ d X = — II I X d X = — '— — h Con. 

But at A, V — and x is sl definite quantity G A = a ; we have, 
therefore, 

= — ^-^ + Con., and 

or the \elocity itself 

V = V fi {a" — x'). 

"When the body arrives dbi C, x = and ^; is a maximum, and 
its value is then 

v=^c — Viia^^=za Vfi. 

Upon the other side of (7, v gradually decreases, and at the dis- 
tance x ~ C B = — a from C it becomes again = ; the body 
then returns with an increasing velocity to C. This return takes 
place in accordance with exactly the same law as the first motion ; 
at (7, V = — c, ana. at A, v = 0. Thus the motion repeats itself in 
the space A B = 2 a, which for this reason is called the amplitude 
of the oscillations (Fr. amplitude des oscillations ; Ger. die doppelte 
Schwingungsweite). 

(§ 2.) The time in which the oscillating body describes a certain 
space A M = a:,. Fig. 875, can be determined in the following 
manner. If in the element d t of the time the element of the path 

M N = d Xy — — d xi^ described, we have (§ 20, I) 

d Xy — V d t, I.E. d x = — Vfjb {d^ — x') d t, 

and, therefore, inversely 

dx 
dt — 



Now if we describe upon A B, with a radius C A = C B = a, 

a circle A D B, Va^ — x^ will be represented by the ordinate M 
= y, and, therefore, we will have 

Vfx. y 
If we put the arc D 0, corresponding to the abscissa C M = x^ 
equal to s, and its differential Q = — d s, we have, in conse- 
quence of the similarity of the triangles Q R and C M,m 



1044 



GENERAL PRINCIPLES OF MECHANICS. 



[§2. 



which R = — d X, 0Q= — ds, M = y, and C = a, the pro- 
portion 

-r— — ^-, and, therefore, 
d s ^ 



Fig. 875. 
D 



a 




— = — ; hence it follows that 
y a 



dt=.-^ 



ds 



M N P C 



ViJb . a 



and 



t 



-f- 



ds 



+ Con. 



But at the point A, where the motion begins, ^ = and s is 
equal to the quadrant DA = J "H" « ; consequently 



= - - 



^ 7T a 



VfX. 



+ Con., 



a 



and the time required by the body to come from ^ to Jf is 



t 



^na 



1 /n 



aj 



The period of half an oscillation, i.e. the time required by the 
body to pass from the point A to the position of rest C, for which 
s = 0, is . TT 



t = 



2.1 



V^? 



and the period of a complete oscillation, or the time required ta 
describe the whole distance A B = 2 a,i^ 



After the time 






the body has made a double oscillation and returned to the point A, 
The time required by the body to describe the space 2 A B = 
4 « is the same, no matter from what point M we begin to count ; 
for the time in which the body goes from Mto B and back is 

__ _ arc B 

— ^ • — ;= ? 

Vii . a 

and that in which it goes from Mto A and back is 

arc OA 



= 2. 



VfM. a 



consequently the time required to describe the space 2 M B -h 
2 MA is 



§3.] THE THEORY OF OSCILLATION. 1045 

== 2 ^^^ {0 B + A) _ 2.7Ta _ 2jn; 

We see that the period of an oscillation does not depend upon 
the amplitude. If we start from the point C, we can put the time, 
which corresponds to the distance C M = x. 



or, since s=-a sm, 



— 1 



= Vjtt . « COS. {t Vfi). 



X 

t = --= sinr^ -, and inversely 

X = a sin. (t Vfji), and 

- a' [sin. {t VJj,)Y=: Vfi. a ^1 



[sin. {t Vii)]' 



Remaek. — Tlie foregoing theory of oscillation is applicable to the cir- 
cular pendulum G if, Fig. 876, if the arcs in which it oscillates are small. 

At A the acceleration of the point, which is oscil- 
lating in the arc A M B, is 

. , ^^ DA 
p = g sin. AGD= -^^ . g. 

or, since for small displacements we can put D A 
= MA, 

DA 




P = 



MA 



9, 



If we denote C Ahj r and ilf J. by a;, we obtain 

gx 

and by comparing it with the formula jp = (W a?, we find 

g 

Hence the period of an oscillation is 



TV / J. 

~'Yu~'' V ~ (compare § 321). 



V/Li 

(§ 3.) Longitudinal Vibrations. — The most common cause 
of oscillatory motion, which is then called vihration, is the elasti- 
city of bodies. The most simple case is that presented by a rod, 
string or wire C, Fig. 877, stretched by a weight G. If we move 
this weight from its position of rest C a certain distance C A — a 
in the direction of the axis of the string and abandon it to itself 



1046 



GENERAL PRINCIPLES OF MECHANICS. 



[§3. 



then, in consequence of the elasticity of the string, etc., it will be 
raised to C, where it arrives with a velocity c and above which it 
ascends, by virtue of its vis viva, to a point B, from 
which it falls again, etc. When at rest, the weight G 



Fig. 877. 




X 



^ is balanced by the elasticity - F E (see § 204) of the 
rod, and consequently tlie motive force is 



B, P=^jFE 



G ^ 0, or J FE = G. 



But if the weight 6^ is at a lower point N, whose 
distance from C is C N= x, the motive force becomes 



X 



P = ~^ FE - G=^j FE ■\- ^ FE 



G 



FE 
I 



Jb^ 



and if it is at a higher point Q, this force is 



P '^G - 



X 



X 



- FE =^ G 



J FE -\-jFE 



FE 



X. 



If we neglect the mass of the rod, the acceleration, with which 
the weight G returns towards C, is 

P F E 

p ~ -p- 9 — p 7 9 ^' ^^^ consequently we have 

_ FEj 

^~ G T' 

when we put p = fi x and denote the length of the rod by /, its 
cross-section by F and its modulus of elasticity by E. As this 
formula corresponds to the case treated in the foregoing paragraph, 
the period of a simple vibration is 



f = ^. 



V/i 



^ FEg ^g ^ FE' 



If instead of F we substitute the weight of the rod Gx = Fly 

E 
and instead of E the modulus of elasticity L = — , expressed in 

units of length, we obtain 



Vg 



G 
G~L' 



§4] THE THEORY OF OSCILLATION. 104? 

If, on the contrary, we observe the period t of the simple vibra- 
tions, we can calculate the modulus of elasticity by putting 

„_7r^ Gl T -^'^ ^ 

J^ — — 12 • ~TT or X/ — -— . -pzT. 

g f F cjfG, 

These formulas also hold good, when the vibrations of the rod 
are produced by simply attaching the weight (at B) ; in this case 
the semi-amplitude on each side of O is 

^'-^^^^ -Ye ^' 

while in the other case we assumed a <^ X. 

A complete vibration is a double oscillation. — [Tr.] 

Example. — If an iron wire 20 feet long and 0,1 inch thick is put in 
longitudinal vibration by a weight G = 100 pounds and if the period of 
a complete vibration is ^ of second, we have t = -ji^, and consequently the 
modulus of elasticity 

E = 0,031 . 7t' . 18^ . i?^-f4--— = 0,031 . 800000 . 18^ . tt 

= 24800 . 324 . TT = 25000000 pounds. 

(§ 4.) The foregoing formula is also applicable to the case, 
where the weight acts by compression upon a stiff prismatical rod. 
It also holds good, when the weight applied at the end of the rod 
has an initial velocity v. According to the principle of mechanical 
effect, when the height of fall of G is h, we have 

Gh -h G^ = j FE. \ = :^ . }i\ and, therefore, 
2 g I 2 2 1 



FE "^ \FEl FE ' 2g 

After the weight G has described this space, it has lost all its 

velocity, and in consequence of the elasticity it rises again to Ay 

where it arrives with the velocity v. In consequence of its vis 

v^ 
viva G ^—, it compresses the rod and rises to a height h^ before 

returning and beginning a new vibration. For this second dis- 
tance we have 

v^ F E 

G ^r—=Ghi-{- ^ , hi^, and, therefore, 
2 g 2 1 

, _ _ _^ i/T^lX ^'^ ^' 

' ~ FE ^ ^ \Fe) "^ FE'2g 
By adding h and hi we obtain the total amplitude of the 
vibration 

7 7 o A/lGTyT^GTir 



1048 GENERAL PRINCIPLES OF MECHANICS. [§ & 

Jience the simple displacement is 



= i/{S) 



a 



FEl ' FE ' %g 

F E 
Since in this case also p — ^ g x = fi x, we have as above 

for the period of an oscillation or simple vibration 

If the initial velocity v of the weight Gi is caused by a falling 

weight G (Fig. 878), we have the case treated in § 348. If the 

weight G strikes with the velocity c, and if we suppose 

Fig. 878. ^j^^ impact to be inelastic, we have the initial velocity of 

4 (^ + G, _ Gc 

'^ - G + G,' 
hence the maximum displacement is 



Gl 



^^/p + ^o^, ^(^"i 



FE I {G + G,)FE' 2/ 
and the period of a simple vibration is 



GxQ" ^9 FE ' 

B The elements of the rod also participate in the vibra- 

tions of 6^ or 6^ 4- 6^1, but their amplitude decreases as 
the position of the element approaches the point of suspension. 
For an element Ci, Fig. 877, situated at a distance C^ — x from 
the point of suspension, the amplitude is 

X 

y ^ ~a\ 

while the period of its vibration is the same as that of G ; for it 
does not depend upon y or a. Hence the vibrations of all the ele- 
ments of the rod are isochronous, but their amplitudes decrease 
gradually from C towards 0. 

§ 5. Transverse Vibrations. — The elasticity of flexure and 
of torsion cause vibrations of the same nature as those just treated. 
If a rod or spring C (Fig. 879) is fixed at one end and deflected 
at the other (7 by a weight G, we have, according to § 217, the 
deflection 



HC = a = 



3 WE' 



§5.] 



THE THEORY OF OSCILLATION. 



1049 



inversely the force, with which the rod is bent, is 



Fig. 879. 



P = 



3 WE a 



Now if this force is re- 
placed by a weight G, at- 
tached at C, and if a is in- 
creased or diminished a dis- 
tance C A = C B = X, we 
have the force, with which 
the rod will be driven back to its position of rest by its elasticity 




3 WE {a-\-x) 3 WE (a + x) 3 W E 



G 



a = 



ZWE 



-x\ 



V - ^3 ^3 -- J. 

hence the acceleration is, when we consider the mass of G alone, 
P 3 WE 



^ 



Gl 
3 WE 
GV 



g X, and, since ^ = fjb x, 



9* 



The relation between p and x allows us to employ the formulas 
of (§ 2), consequently the period of an oscillation or simple vibra- 
tion is , __ _7T^ __ _7T_ 4 / G r 

If the rod H 0, Fig. 880, is supported at both ends and loaded 
in the middle C with a weight G, we have, according to § 217, 

^ __P/ 
Fig. 880. " ^ 

HBO 



a = 




48 WE' 
and, therefore, the duration 
of a simple vibration 



t 



- ^. /, 



Gf 



^g ' 48 WE' 

If we take the weight G^ of the rod into consideration, we must 
substitute in the first case. Fig. 879, instead o^i G, G ^r \ G„ and 
in the second case, Fig. 880, instead of 6^, 6^ + ^ G^, 

From the observed duration of an oscillation or simple vibra- 
tion we can calculate the modulus of elasticity, in the first case by 
the formula „ l^\ 1^ -^ \^A 7> 

It/ \t7w~) ^ 



E 



or, if fi = - denotes the number of simple vibrations per second. 



E 



(TT nf ( 



G + \ 



dgW 



1050 



GENERAL PRINCIPLES OF MECHANICS. 



[§6. 



Example. — A pine rod 1 centimeter square is supported at two points 
100 centimeters apart, and its centre is deflected a distance a = 3,2 centi- 
meters by a weight G =. 1,37 kilograms. According to this experiment 
the modulus of elasticity of pine is 

P P 1,37 . 1000000 ^ ^^„, , ., 

= 107031 kilograms, 



E = 



48 Wa 48 . yV • ^,2 
while in the table on page 370 we find E = 110000. 

The rod was then firmly fixed at one end, was loaded at the other with 
a weight G = 0,31, and put in vibration. It was found that the number 
of simple vibrations in 35 seconds was 100. The weight of the rod wai 
G^ = 0,044 kilograms ; hence G + I G^ = 0,331 kilograms and 



E 



{tf-y dgW J V0,35/'9 



321000 



--= 80,57 



dg W 
128100 
981 ~ 



881 . -,\ 



105260 kilograms. 



or about the same value of E as was found by the experiment upon flexure. 



§ 6. Vibrations Due to Torsion. — The formula t 



7T 



Vll 



can 



Fig. 881. 

1 



also be applied to the torsion halance or torsion rod (Fr. balance de 
torsion ; Ger. Torsionspendel), i.e. to a thread or rod D 0, Fig. 
881, oscillating about its axis, in consequence of its torsion. Gen- 
erally the rod is provided with a loaded 
arm C C^, by means of which the origi- 
nal torsion of the thread is produced, 
by bringing this arm from its position 
of rest G C^ into the position A A^. 
The torsion drives the arm back to 
C Ci, and the latter, by virtue of its in- 
ertia, moves further on until it comes 
into the position B B^, from which it 
returns to O C^ and A Ai, etc. We 
found previously (§ 262) the moment 
of torsion of a prismatic body to be 

a Wc 




P a = 



I 



we know, therefore, from this formula, that it is inversely propor- 
tional to the length D = I of the rod and directly proportional 

to the angle of torsion M D G = a; now if — Ic^ is the moment of 

inertia of the arm CD Ci, ^ — is the mass M reduced to the ends 
G and Gi of the arm, and the acceleration of this point is 



§7.] 



THE THEORY OF OSCILLATION. 



1051 



P 



aWc ¥ a aaW G g 



I a ' a^ g G ¥ I 

If we denote the arc G M — a a, corresponding to the length 
of the arm D A = I) G = a and to the variable angle of displace- 
ment G D M =^ ^? by X, we obtain the expression 
__ WGg 



P = 



^ 



Qiei 

_ WGg 



X, and we can again put p ^ fix, or 



G¥l' 
The period of an oscillation or simple yibration is, therefore, 

^ JLa/~gW1 



t = - 



Vfi Vg' W G 
no matter whether the amplitude A GB =^ Ai GiBiis large or small. 
Inversely, we have 

WG= ^,G¥l, 
gf 

and, therefore, the moment of torsion 
F a = -—^ . a G k\ 

Remark. — The above formulas for the vibrations produced by the 
elasticity of rigid bodies are not correct unless the displacement during 
the vibration is within the limit of elasticity. Great care should be 
taken to avoid as much as possible vibrations in the various parts of 
machines ; for the energy expended upon them is lost to the machine. 
For this reason the parts should be united to each other with precision, 
and what is known as lost motion is to be avoided, as it gives rise to con- 
cussions and vibrations. 

§ 7. Density cf the Earth.— The 

theory of the torsion-rod can be directly 
applied to the determination of the mean 
heaviness or specific gravity e of the earth. 
If we cause a heavy sphere K to approach 
the weight G, which is fastened upon the 
end of the arm ADA,, Eig. 882, the 
latter will be attracted towards the former 
a certain distance A M = x ; the attrac- 
tion R oi K balances the force of torsion 
P, when G occupies the position Jf ; one 
of the above forces can, therefore, be de- 
termined from the other. Now if we re- 
move the heavy sphere K and allow the 



Fig. 882. 




1052 GENERAL PRINCIPLES OF MECHANICS. [§ 7. 

torsion-rod to vibrate, we can observe the period of the vibrations, 
and from it we can calculate the force of torsion. According to 
the foregoing paragraph, the period of a simple vibration is 



TT p ^ force of torsion F a^ 

t = — — , fji = - and^ = 



ff^ 



Vfjb ^ mass of torsion-rod G ¥ 

when G h^ denotes the moment of inertia and a the length of the 

arm of the torsion-rod ; inversely, the twisting or attractive force is 

p _ a¥ p _ \iG¥ X __ _7T^ G¥ X _ 7^ G¥ a 

~ g a' ~ go" ~ g f a" ~ gf a ' 

and the moment of torsion corresponding to the angle of torsion a is 

_2 

P a ^^ — ^ . a G ¥. 

Now if the forces, with which the bodies attract each other, vary 
directly as i\\Q\v masses and inversely as the squares of their distances 
(see § 302, Example 3), we can compare the attraction P, exerted 
upon the body by K, with the weight Q of the small body which is 
placed upon the torsion rod; for the weight is the measure of 
attractive force of the earth ; thus we obtain 

P _ K:s' 
Q ~ E: r'' 
in which s denotes the distance M K of the centres of the two 
masses G and K from each other, r the radius of the earth and E 
its weight. If we solve the above equation, we obtain the latter 
weight ^ K Q r^ 

and if we substitute ^ = f tt r^ , e y, we have the mean heaviness 

of the earth . 

_ _ dE _ S EQr'' _ ^KQ _ ^KQ g f a' 

^' — ^ ^ - 4: 7T r' ~ 4: n P r' s' ~ 4: n P r s' ~ 4.n r s' ' n' G ¥ x' 

or if we introduce the length of the second pendulum I = -^ (see 

S323), ^Klf Qa" 

y^-^y- 4^^rxs''' G¥' 
hence the mean specific gravity of the earth is 

_ 3 Klf Qa " 
^ 4iTT r X s^' Glc'y 
If we put approximatively G ¥ = Q «% we obtain more simply 

Elf 



s =z ^ 



A 2 * 

^ 7T r X s y 



Cavendish found in the first place with the torsion rod, or 
Coulomb's torsion balance, as it is called, s = 5,48 ;. or, according to 
Hutton's revision, e = 5,42. 



§8.] THE THEORY OF OSCILLATION. 1053 

Eeich found afterwards, with the aid of the mirror apparatus of 
.Gauss and Poggendorff, e = 6,43. Baily, on the contrary, found 
by experiments upon a larger scale, e = 5,675. 

When Reich repeated his experiments he found € = 5,583. (See 
" Neue Versuche mit Drehwage, Leipzig, 1852.") The mean 
density of the earth is, therefore, according to these experiments, 
about equal to that of specular iron. 

Remark. — The following works may be consulted in reference to the 
manner in which the density of the earth was determined : '' Gehler's 
physikal. Worterbuch," Bd. Ill ; the treatise of Reich " Yersuche iiber die 
mittlere Dichtigkeit der Erde, Freiberg, 1838 ;" and that by Baily, " Ex- 
periments with the Torsion Rod for Determining of the Mean Density of 
the Earth, London, 1843." 

§ 8. Magnetic Needle. — The torsion-balance may also be 
employed to find the directing force or the moment of rotation of a 
magnet or of a magnetic needle (Fr. aiguile aimantee ; Ger. Magnet- 
nadel). If we replace the transverse arm of the balance by a 
magnetic needle or by a bar magnet M D M^, Fig. 883, it will as- 
sume a position in which the directing force is 
balanced by the twisting force. If the non-mag- 
netic arm, when at rest in ^ ^i, forms an angle 
A D N = a with the magnetic meridian N S, 
and if the bar magnet M Mi assumes such a posi- 
tion that its axis forms an angle M D N = S 
with the meridian iV^ /S> we have R^ = R sin. d, in 
which formula Rx denotes the component of the 
directing force R, which is parallel to N 8. This 
component tends to turn the needle, and is bal- 
anced by the force of torsion. The latter force 
P, on the contrary, is proportional to the angle of torsion MD A — 
a — 6, and we can, therefore, put 

P = P,{a-d)', 
hence we have R sin. 6 = P [a — 6), and consequently 

when the variation or angle of deviation 6 is small. 

Now according to the foregoing paragraph the force of torsion 
is expressed by the formula 

P - — ^^' ^ - ^ Gh' a{a - 6) _ 7T^ Gk' {a - 6) 
~ gf a' ~if' a' ~ gf a 

and we can calculate from the period t of an oscillation, etc., of the 




1054 GENERAL PRINCIPLES OF MECHANICS. [§ 9. 



non-magnetic torsion-rod the directive force of the magnetic needle 
bj the formula 



^ ' g f a 



The moment of this force, when we assume that it is applied at 
a distance D M = a from the axis of rotation and when the varia- 
tion is if D iV = c5, is Ri a = R a sin. cJ, approximatively, for 
small variations, 



rr^ 



= R a d = la - 6) , -~ . G k\ 

^ ' gt\ 

This moment {R a sin. d) is a maximum and — R a for 
sm. 6 = 1, I.E., when the magnetic needle is at right angles to the 
magnetic meridian, and, on the contrary, a minimum and =: 0, 
when cJ = 0, i.e., when the axis of the maguet needle coincides with 
the magnetic meridian. 

§ 9. Magnetism. — Since the directive force of the magnetic 
needle causes no pressure upon the axis, I.E., the needle has no 
tendency to move forward, but only a tendency to turn, when its 
axis does not coincide with the magnetic meridian, it follows that 

the entire action of the earth upon the magnet must consist of a 

7? 7? 

couple — , — — , the maximum moment of which is R a. Now 

since every couple — , ^ can be replaced by an infinite number 

of other couples \~^-> ~~ ~^\\ "^? — ^" I? ^^c, whose moments 

R «, Rx aiy R^i a<i, etc., are equal to each other, it follows that nei- 
ther R nor a, I.E., neither the directive force nor the point of appli- 
cation, but only the moment R a is determined. This twisting 
moment depends, in addition, upon two factors, jWi and S, ii^ corre- 
sponding to the magnetism of the earth and S to that of the bar or 
needle ; hence we can put 

R = jji^ S and R a = fii S a. 
The measure i^i of the magnetism of the earth for a needle 
vibrating horizontally (the case under consideration) is only the 
horizontal component of the intensity fi of the entire magnetism 
of the earth ; for the vertical component fio is counteracted by the 
support of the needle. If t is the angle of dip or inclination or the 
angle formed by the magnetic axis of the earth with the horizon, 
we have the horizontal component 

fii = 11 COS. i; 
on the contrary, the vertical one 

l^i — ^ sin. L, 



§10.] 



THE THEORY OF OSCILLATION. 



1055 



and, finally, the twisting moment of a magnetic needle is 

R a sin. 6 — ^ cos. t . S a sin. o, 
tiie maximum value of which is 

R a =^ [i S a COS. i, 

§ 10. Oscillations of a Magnetic Needle.— We can calcu- 
late the moment of rotation of a magnetic needle from the period 
of its oscillations. If we move the suspended needle M D Jf„ Fig. 
884, from its position of rest, where the force of torsion and tlie 

directive force of the magnet are in equilibri- 
um, so that its new position shall make a small 
angle M D C = (f) with its former one, either 
the magnetic directing force R is increased by 
R and the force of torsion Pj is diminished 
by Pi 0, or the reverse takes place ; in either 
case their resultant 

{R + A) </> 
or its moment 

(R ^ P,) (P a = {R -\- P,) X 
drives the needle back to its position of resi 
If G F is the moment of inertia of the needle, the acceleration, 
corresponding to this force, is 

(R + P^ ax 




G¥ 



if we put it = [ix, we obtain 

' (R+P^ 



G¥ 
and the period of an oscillation is 



) 



ag, 



t = 



ar 



^^ a/I 

V^ "'^{R-¥P,)ag 



G¥ 



P 6 

or, if V denotes the ratio ^ 



R 



a — 6 



of the force of torsion to the 



magnetic force, 



G¥ 



'" i^^ (1 + v)Ra 



If we have found t by observation, we can find by inversion the 
moment of rotation of the needle, which is 



1056 GENERAL PEINCIPLBS OF MECHANICS. [§ 11. 

Jh d = ;;; . 



gf 1 A-v 

If the force of torsion is small, I.E., if the position of repose 
nearly coincides with that of the magnetic meridian, we gb^q 
neglect v and put 

t = —- y -^ — and 



Vg 



R a 



We can also substitute for R a its value, which has been given 
auove, and express the moment of rotation by the formula 

11 S a COS. i = — 77, . G Ic^. 

For a di2oping needle, which oscillates in the plane of the mag- 
netic meridian, we Iiave, on the contrary, 

11 8 a = —^. G tc", 

and for a needle, whose axis lies in the magnetic meridian and 
which, therefore, tends to place itself in a vertical position we have 

iJb 8 a sin. I = —-, . G Ic^. 
gf 

In the formula ii 8 a cos. t = — ^ , G k'', fi 8 a cos. i is a product 

of four factors ; however, since the inclination c can be determined 
by observing a magnetic needle, and since 8 a cannot be decom- 
posed into two definite factors, we have to only resolve the product 
fj> 8 a into the factors ft and 8 a. How this can be done by ob- 
serving the declination of the needle will be shown in the sequel. 

§ 11. Lavr of Magnetic Attraction. — The forces, with which 
the opposite poles of two magnets attract and the similar poles 
repel each other, are inversely proj^ortiojial to the squares of their 
distances from each other. We can convince ourselves very easily 
of this fact by observing a small magnetic needle, which has been 
set in oscillation near a large bar magnet. The bar magnet is 
placed in a horizontal position and in the plane of the magnetic 
meridian, its north pole being directed to the north and the south 
pole towards the south ; we then place a small variation compass 
in the prolongation of the axis of the bar magnet. If the distance 
s of the pivot of the needle from one pole of the bar magnet is 



§13.] THE THEORY OF OSCILLATION. 1057 

much less than its distance from the other pole, we can disregard 

the action of the latter upon the needle and we can assume that, 

in consequence of the action of the nearer pole, the coefficient ^ of 

the magnetic force of the earth is increased a certain amount k^ or 

K^, If the period of the oscillations of the needle is = t, when the 

bar magnet is removed, and, on the contrary, if it is = t^, when 

the nearer pole of the bar magnet is at a distance Si from the pivot 

of the needle, and = t^, when the latter distance is = s^, we have 

tt' tt' 7r' 

^ gt ^ ' gf ^ ' gti 

whence we obtain by division 

jLfci 4-^1 f T A*i + Ko f 
— = —^ and = —2 ; 

resolving the last two equations, we obtain 

«i = ( — p-^j /^i and K, = I — p-^j ^1, and, finally, 

_ f - t;' ^ f - 1? 

or, if we substitute instead of t, U and t^ the number of oscillations 

60" 60" ^ 60" 

n = -T-, ni = — - and oh = —r-, 

Ki'. K^ = n^ — n" : n^ — n^- 
If the action of the bar magnet upon the magnetic needle is 
inversely proportional to the square of the distance, we must have 

also 

«j : /Cg = 5/ : Si^f and therefore 

?^l' — n"" _ 52' 
n^'' — n'' ~ Si" 
which is confirmed by the observations. 

§ 12. The actions of a bar magnet N S upon a magnetic needle 
n s are simplest, when the bar magnet is placed at right-angles to 
the magnetic meridian in such a manner that the pivot of the 
compass ?^ s, Fig. 885, lies either in the prolongation of JV S or in the 
line which is perpendicular to JV S, Fig. 886, and passes through 
its middle C. If for the present we put the force, with which a 
pole of W S acts upon a pole of n s, when their distance apart is 
unity, = K, we have in the first case. Fig. 885, when a denotes the 
length iV S and e the distance C d between the centres C and d 
of the two bodies iV S and n s, the force, with which the north 
pole n is attracted by S, 

67 



1058 



GENERAL PRINCIPLES OF MECHANICS. 



[§ia 



P = r==, approximatiyeiy = ---,, 

and the force, with which n is repelled by iV, is 



Fig. 885. 



Fig. 886. 





P = 



K 



K 



hence the resultant of P and P^ is 

^ {e H- I ay -{e - i a)] ^ 

- 2ae X 

~ (e + i ay (e - 1 af 
or, if ^ 0^ is small compared to e, 

_ 2 aeK __ 2aX 

In like manner we iind the resultant of the attraction and 
repulsion of the south pole s 

2aX 



Q = 



,3 ? 



hence the moment of the couple, formed by these forces, is 

^ J _ 2 al K 

when I denotes the distance between the two poles of the needle. 

For the second case (Fig. 886), on the contrary, the attraction 
and repulsion at s are 



P^ = 



K 

K 



K 
K 



-, and those at n are 



Sn^ N n^ 



hence the resultants are 



13] 



THE THEORY OF OSCILLATION. 



1059 






a P 

Ns 



a K , ^ a K 
and (4 = 



Now if h a and I I are considerably smaller than e, we can sub- 
stitute for J\^ 5 = '8 s and N n — 8 n the mean value N d = 8 d 



and for the latter the approximate value C d = e; thus we obtain 

Q = Q. = ^, 

and, therefore, the moment of the couple, formed by Q and Qi, 

^ _ al K 

^ ^ - ~e~' 
I.E., it IS one-half as great as in the foregoing case, a result which 
is perfectly corroborated by observation. 

But the force K is itself a product of the intensity tc of the 

magnetism of 71 s and the intensity 8 of iV^ 8, i.e., IC — k 8; hence 
we have in the first case 

^ 2 fi 8 a T ,1 T ^ «■• 8 a 

= ^ — , and m the second case Q = — ^. 

^ e e 

§ 13. Determination of the Magnetism of the Earth. — 

Jf in both the above-mentioned cases the magnetic needle n s is 
abandoned to the action of the larger magnet, the former will 

assume a new position n s, Fig. 
887, in which the force Q, with 
which the bar magnet acts upon 
the needle, is balanced by the 
force E, due to the magnetism of 
the earth. If 6 is the variation 
]^ d n = 8 d s of the needle from 
the magnetic meridian, we have 
for the components of Q and B, 
which balance each other, 
Q^ z= Q COS. 6 and R^ =. R sin. 6 ; 
Q COS. 6 =: R sin. 6 and 




hence 



tang. 6 = 



Q 

R' 



or, if we put, according to the last paragraph, either 

^ 2 K 8 a ^ K 8a 
Q = — :j— ot Q = —J-, 



and, according to § 9 of the Appendix, R = jt*, «, we obtain either 



1060 GENERAL PRINCIPLES OF MECHANICS. [§13. 

„ 2 K S a % Sa ^ ' „ S a 
tana, o =z = -, or tmiq, 6 = — 5. 

By inversion we obtain the ratio of the magnetic moment of the 
bar to the intensity of the magnetism of the earth; for in the first 
case we have 

— = ^ e^ tang. c5, and in the second case, — = e^ tang. 6. 

By observing the period of the oscillations of the bar magnet, 
we obtain (according to § 10) the product 

a^ 8 a = —^(jr'k''\ 

gt 

by combining the two equations, we deduce the magnetic moment 
of the bar, which is 

7r 



either 8a — 777= V\ G ¥ e^ tang. 6 



n 



or 8a — 7^7= ^G- ^' e^ tang. 6, 

and the measure of the horizontal component of the magnetism of 
the earth, which is either 

— ^ i/ ^ ^ ^^ cotang. 6 _ ^ . /6^ F cotang, d 

the first formula being applicable to the case represented in Fig, 
885, and the second to the case represented in Fig. 886. If we 
divide by the cosine of the angle of dip or inclination {i\ we obtain 
the total intensity of the magnetism of the earth 

\h — . 

COS. I 

In order to obtain a clear idea of the coefficient or measure \i of 
the magnetism of the earth, we must put in the formulas 

R a ^ II 8 a and Q I = ^ — , a = I = e = 1, 

e 

and also k = 8 =1', thus we obtain E a = fi and § Z = 1 ; hence 

1) the measure /x of the intensity of the magnetism of the earth 
is that moment, with which a magnetic needle, whose magnetic mo- 
ment is = unity, will be turned by the magnetism of the earth ; and 

2) the magnetic moment of a magnetic needle is = unity when 
that needle communicates to another similar and equally powerful 
magnetic needle, placed in the position represented in Fig. 886 at 
the unit of distance from it, a moment = unity (1 millimeter-milli- 



§14.] 



THE THEORY OF OSCILLATION. 



1061 



gram). According to Weber, if the acceleration of gravity were 
1 millimeter, we would have 

in Gottingen fi = 1,774 millimeter-milligrams, 
in Munich fi = 1,905 " " 

in Milan fi = 2,018 " ". 

but, since the acceleration of gravity in Central Europe is 9810 
millimeters, the true values are t^9810 = 99 times less. 

Remakk. — We would recommend to those who wish to make a more 
extended study of magnetism, besides Miiller-Pouillet's "Lehrbuch der 
Physik ;" Lamont's " Handbuch des Erdmagnetismus" (Berlin, 1849), and 
Gauss and Weber's " Resultate aus den Beobachtungen des magnetischen 
Vereins," Gottingen and Leipzig, 1837 to 1843 ; also the " Experimental- 
physik" of Quintus Julius, and Mousson's *' Physik auf Grundlage der 
Erfahrung," etc. 

§ 14. Waves. — In discussing the longitudinal and transverse 
vibrations of prismatical bodies, we have heretofore (§ 3, 4 and 5) 
neglected the mass of these bodies and consider.ed only that of the 
weight, which produced the strain in the bodies. Hereafter, on the 
contrary, we will not consider any such weight, but suppose that 
the body is put in vibration by a sudden blow or by a force, which 
acts for an instant only ; we must, therefore, take into account the 
inertia of the vibrating body alone. As the most simple case is 
that offered by longitudinal vibrations, we will, therefore, treat that 
first. 

From what precedes, we know that all the parts of a prismatical 




rod B M^, Fig. 888, are put in vibration, when this body is extended 
or compressed by a force P, acting in the direction of its axis. Not 



1062 GENERAL PRINCIPLES OF MECHANICS. [3 15. 

only the element M at the end, but also every other element 
Jfi, M.2, il/3 . . . . of the rod vibrates back and forth in a certain 
space B D, B^ D^, B^ Dl . . , which is called the amplitude of tlie 
vibration ; we can also assume, when the rod is very long, that this 
space is the same for all the elements. Although the time in 
which an element makes a vibration is the same for all parts of the 
rod, we cannot, therefore, assume that all these elements J^/, M^, M., 
etc., are simultaneously in the same phase of motion, e.g., that they 
are all at the same time m the middle of a vibration, but we should 
rather suppose that time will be required to communicate the mo- 
tion proceeding from Mto the succeeding elements, and that the 
farther an element is situated from the origin F of the motion, the 
later it will enter upon the same phase of motion. It is, therefore, 
possible that at the instant, when the element M has made a com- 
plete vibration B D forward and back, the element M^ has made 
but one-half of its forward movement and has arrived at 63, and 
that the element M^, is just beginning a vibration. The latter will 
therefore vibrate isochronously with M. The velocity with which 
the same phase of motion advances in the body is called the velocity 
of propagation (Fr. vitesse de propagation; Ger. Fortpflanzungs- 
geschwindigkeit) of the vibrations of the body. The aggregate of 
all those elements between M and i/4, which are in the different 
phases of a complete vibration or which are included between two 
elements M and J/4, which are in the same phase, are called a ivave 
(Fr. ondulation ; Ger. Welle) of the vibrating body, and the dis- 
tance M M4, is called the length of the wave. A wave consists of a 
back part B D^_ which contains the returning elements, such as 
Jfi, J/c, .... and of the wave front Dc^ B^, which comprehends the 
advancing elements M^, M^ . . .; B D., is also called the rarefied 
and Do B^ the condensed portion, since all the elements in B D.2 are 
extended and those in D^ B^ are compressed. 

§ 15. The phccses of the motion and of the velocity in a wave can 
be very well represented by serpentine lines, such as F Ci Gc^ C3 H^ 
and B M, D, ^^3 B^, Fig. 889, I and II. At the moment when M 
begins a new vibration at B, its displacement is a maximum and 
its velocity is = ; at the same time M^ is in the position of rest, 
and consequently its displacement is = and its velocity is a max- 
imum ; both of these facts are shown by the above curves ; for the 
first curve (that of the displacement) (I) passes at .5 at a distance 
equal to the amplitude B F = B (7 above the axis B B^ and cuts 



§15.] 



THE THEORY OF OSCILLATION. 



1063 



this axis at Ci, while, on the contrary, the second curve (that of the 
velocity) (II) cuts the axis at B and at (7i passes at a distance 
Ci i/"i, equal to the maximum velocity, above the axis. At the 
same moment the element M.^ is upon the other side of the position 
of rest C.2 and at the maximum distance from it, and its velocity, 
like that of J/, is = ; this is also shown by the two curves ; for 
one passes at A at a distance equal to the amplitude D.^ G^ below 
the axis, and the other cuts it at that point, so that the ordinate 
wiiich corresponds to the velocity is = 0. In like manner the 
phases of the motions and of the velocities of the elements Jis, M4, 
etc., are represented by these curves. Since, e.g., the first curve 
cuts the axis at C^ and the second passes below that point at a dis- 
tance equal to the maximum value C^ N-i, we know that the ele- 
ment M^ at this moment passes through the position of rest with 
the maximum velocity in the positive direction. If we wish to 
know the phase of the motion of any other element M.2, situated 
between M, M^, M4, etc., at the moment when the element if« be- 
gins a new vibration, we have only to let fall from it a perpendicu- 
lar upon the corresponding curve. The portion R S of this per- 
pendicular lying between the curve and the axis corresponds to the 
displacement of this element, and the portion T U, between the 
second curve and its axis, gives its velocity. Since both ordinates 
are directed downwards, we know that both the displacement and 
the velocity are positive, i.e. their direction is that of the velocity 
of propagation. 

If the element M were at Z>, i.e. about to begin its return mo- 
tion, the displacements of the other elements of the wave would be 
represented by the dotted hne / Ci K-2 C^ L^, and their velocities 

Fig. 889. 

Ma Mn 



M 



Ttfi 



Ms 



M4 




1064 



GENERAL PRINCIPLES OF MECHANICS. 



[§16. 



by the ordinates of the dotted curve D 0, B^ ft D^, The period 
of a double oscillation or that of a complete vibration, i.e. the time 
t, in which the space B I) -\- I) Bis described, is equal to the time 
in which a vibration is propagated through the length i¥ M^ = I 
of a wave ; if, therefore, c is the velocity of propagation, we have 
the total length of the wave 

BB, = l = c.2t = 2ct. 
The length of the back part of the wave is 

B n, = I, ^ B B, + B, D, = c t + X, 
and that of the wave front is 

n, B^ = U = D, D^- B,D^=^ ct - X, 

in which X denotes the amplitude of a vibration, 

Remakk. — The phenomena accompanying the interference of waves can 
be shown by the aid of the curves of vibration. Let us consider two sys- 
tems of equal waves, which are advancing in opposite directions, and let 
A B G D E and F G H I K^ Fig. 890, be the curves, whose ordinate 5 rap- 

FiG. 890. 




fesent the displacements. The displacements of an element, which be- 
longs to two waves, produce a mean displacement, which is determined in 
exactly the same manner as the resultant of two motions (see § 28), that is, 
by adding algebraically the two component displacements. Hence at the 
two points if and iV, where the two curves meet each other, the ordinates 
are doubled, and, on the contrary, at the points and Q^ where the curves 
pass at equal distances from, but on opposite sides of the axis A E^ the or- 
dinates cancel each other, and the resultant of the two wave curves is a 
third curve F B B H 8 B Q K, whose ordinates give the displacements 
of all the elements in the axis A E. While the two systems of waves AB G 
and F G Hare moving towards each other, the j)osition of the wave-curve 
F B B 0, etc., of course changes ; but it is easy to understand that the 
points of no motion and Q do not change ; for the ordinates of these 
points of the two component curves are always equal and opposite. These 
points are called the nodes. 

(§ 16.) Velocity of Propagation. —The velocity of propaga- 
tion of waves can be determined in the following manner. Let us 
imagine the vibrating body B 0, Fig. 891, to be composed of an 
Infinite number of elements, the cross-section of each being A and 



§16] THE THEORY OF OSCILLATION, 1065 

its length B G — C D =^ d x, and let us assume that the phase of 
the motion of an element B C = A d x is propagated completely 

to the following C D — A d x 

Fig. 891. in the elementary time d t, oi 

>g ^ J, iijfNN o that the phases of the motion 



are propagated in the direc- 
tion of the axis of the body 

d X 
with the velocity c = -^. Let us assume that the elements B C 

and G D oscillate from C to iV^ in the time t, and thus come into 
the position M N = d x^ and N = d Xc„ and let us denote the 
corresponding displacement G Nhj y. If the surface of separation 
of the two elements, which before d t seconds was at iVj, comes 
after d t seconds to N^, the corresponding spaces described by these 
elements are 

N JSTi = d yi and N N^ = d y^y 
and their velocities are 

dyx T dv^ 

hence the retardation is 

v^ — V _ dyx — dy<i 



V = 



dt dt 



Since d t seconds before the moment, when the elements B G 
and G D occupied the positions M N and N 0, N^ was in the same 
phase as now is, we have G JV^ = D 0; and since d t seconds 
later iVg is in the same phase as M, it follows also that G N^ = B M, 
From these two equations we obtain 

N,0 = DO-DN,=DO- {GN, - CD) = GD and 
MN,= GN,- GM= GN,- (B M - B G) =: B G; hence 
NN,= dy, ^N, - N0= GD - N0 = dx- dx.^idi 
NN^=dy, = MN^ - MN= B G - M N =^ dx- dx,. 
The element d y oi the space is equal to the compression 
d X — d Xi of the element JV 0, and the element d y.^ of the space 
is equal to the compression d x — d x^ of the element M N. If 
we denote by E the modulus of elasticity of the vibrating rod, the 
strains of the elements if i\^and N produced by this compression 

are ^ ldx-J^\ j, ^ = P^ . A E ^r.d. 

\ dx I d X 

S, = (^1^) AB = i-^.AB. 
\ dx 1 dx 



1066 GENERAL PRINCIPLES OF MECHANICS. [§ 16. 

If we subtract the former from the latter, we obtain the retard- 
ing force 

P = ^, _ ^, = (Ijt^yi) J X 

\ a X / 
If y is the heaviness of the elements B C, C D, etc., of the rod, or 

A d X . y the weie'ht, and ^^ — the mass of such an element, its 

acceleration at N^ is 

^i_ ^ j dyx ~dy A ^ ^ g ^ g_E^ dy,-dy^ ^ 

■^ M \ dx I ' A d X .y y ' d x^ ' 

equating the two values of j9, we obtain 

dy^-p^ ^ g_E _ dy,-^^^ ^^^^^^ 

d t y d X' 

dx^ gE ^ qE 
-ji:r = — -, ore' ^ -— ; 
dt y y 

hence the velocity of propagation of the ivaves (velocity of sound) is 



^/£ 



E 



Vg L, 

in which formula L denotes the modulus of elasticity expressed in 
units of length. 

Example. — If we assume the modulus of elasticity of spruce wood to be 
E — 1870000 pounds and the weight of a cubic foot of it to be = 30 
pounds, we obtain the velocity of propagation in it 



/ 



144 . 1870000 



. ^ = V 48 . 187000 . g = 17000 feet, 



30 
I.E. about 15 times as great as in air. 

Eemark. — This formula for the velocity of propagation is applicable 
not only to a stretched string, but also to water and to the air. If p de- 
note the pressure of the air upon the unit of surface, we have, according 
to Mariotte's law, the tensions corresponding to the ratios of compression 

pdx pdx pdx _ pdx 

^~ dx^~dx —dy^ ^ ~~ dx^~dx — dy^ 

and, therefore, the motive force upon an element, whose cross-section is J., is 



{dx — d y^ {dx — d y^ ' 

je -7— is a small fraci 
d X 

dx^ and 



d 11 
now since -~- is a small fraction, we can put {dx — dy^ {dx — d y^) = 

0/ Q) 



p— (^yi —dy^)Ap 
dx 



§17.] THE THEORY OF OSCILLATION. 1067 

This expression agrees exactly with the former one when we substitute 
p instead of E; hence the velocity/ of sound in air is 



r y 

When the theory of heat is discussed in the second volume, it will be 
shown that a coefficient must be added to this formula in consequence of 
the change of temperature, which necessarily accompanies the change of 
density of the air. Since the heaviness of the air is proportional to the 
pressure p, they both disappear from the formula and the temperature 
alone remains. We generally assume for air 



c = 333 VI + 0,00367 . r = 1092,5 Vl + 0,00367 . r feet. 
Example. — If (according to the Remark of § 351), when a column of 
water is compressed by a force of 14,7 pounds, its volume is diminished 
0,000050 of its original volume, its modulus of elasticity is 

and the velocity of sound in water is 



,/ 294000.144 ,/, ^ 1693440 

. = |/ 32,2 . —g^^^^ ^ |/ 32,2 . ^^^^ = 4673 feet, 

or about 4,3 times that in air. 

(§ 17,) Period cf a Vibration. — We can now find the period 
of a vibration by obtaining the equation, which expresses the de- 
pendence of the amplitude of the vibration upon the time and 
upon the abscissa x, which determines the position of the vibrating 
element when it is at rest. Now y is certainly a function of t as 
well as oi x\ we can, therefore, put y = (p (t) and y = 'ip (x). 

By differentiating the first equation, we obtain the variable ve- 
locity of vibration dy , , ,. 

and in like manner, by a second differentiation, the corresponding 
acceleration dv , ,,. 

P = Tt = ^' *''' 

in which 0i {t) and 02 {t) denote other functions of t (compare § 19). 
The second function gives the ratio 

S = ^' (^)' 

which determines the strain ; from it we obtain the latter 

8= A E.^ = AB.ip^(x); 
dx 

hence the motive force of the element of the mass dM— Adx-is 

g 



1068 GENERAL PRINCIPLES OF MECHANICS. [§17. 

ax a X 

and the corresponding acceleration is 

in which ipi (x) and V'2 (^) denote other functions of x. 
■ If we equate the two values of ^j, we obtain 

^ ... a E , , . , , a E „ 
(/)jj {t) = - — . V^2 (^)> or, since - — = c , 

02 (0 = c^-tp^ (x). 
The integral of this differential equation is 

y = (f) {t) = ijj (x) = F{c t + x) + f (c t — x), 
in which F and/ are undetermined functions of the quantities con- 
tained in the parentheses ; for 

0, {t) = ^-^j~- = cF,{ct + x) + cf, {c t - x\ 

cj>, (t) = ^-^^Y^ ^e F,(ct ■\- X) ■\- c\f, (ct-x) 
= & \F^ (ct + x) + f% (ct — x)], and 

tp, (x) = ^ ^"^^ ^^^^ - F, (ct + x) - /i (ct-x) and 

^^.(x) = ^^ = F, (c t + X) +f,{ct-x), 

and, therefore, we have really 
02 (t) — c" .i)^ {x). 
Although the function 

y = F {ct + x) + f{ct — x) 
is an indeterminate one, yet, when we have more definite data in 
regard to the vibrating body, it can be employed to determine the 
period of the vibrations. A few examples of how this may be done 
will now be given. 

Remark. — If we eliminate d t from the formulas d y =^ ^ dt and d x = 

T ^ ■, J ' .-, . d y 1) . d y 

c a t, we obtam the expression -^ — = -, or since -^-^ expresses the conden- 

sation a of the vibrating element of the body, we have a = --, the simul- 

c 

taneous condensation at every point of the vibrating rod is proportional to 
the velocity of vibration of that point. 



§18.j THE THEORY OF OSCILLATION. 1069 

(§18.) Detennination of the Modulus of Elasticity. — Let 

us assume that the vibrating body, whose length is I, is fixed at 
toth ends. In this case we have not only for x — 0, but also for 
X — I, y = 0', hence 

F (ct) + f {ct) = ^ndi F {ct + T) +f{ct-l) = 0. 
From the first equation we obtain f = — F, which, when sub- 
stituted in the second equation, gives 

f{ct + I) -f(ct~ I) :=0,l.B.f{ct + I) =f{ct-l), 

or, if we put ct — I = cti, 
f{ct, + 2 1) =f{ct^). 
The function, therefore, assumes the same value when c ti is in- 

2 I 
creased bv 2 / or when the time is increased by ^i = — ; hence the 

c 

period of a comjMe vibration or double oscillation is 

#, = ^^ = 2 / /-^. 
c ^ g E 

If, in the second place, we assume the body to be free at both 
ends, we have for 2: = and x = I, 8 — and 1/^1 (a;) = ; hence 
^1 (P t) -/i {ct)=^ and F^{ct + 1) -f{ct-l) = 0. 
We have, therefore, 
f ^ F, and/, (ct + 1) ^f [ct - I) , or/, {ct^ + 2Q =^f{ct,), 
and consequently the period of a complete vibration is 

c 
If the body is free at one end and fixed at the other, we have for 
a; = 0, «/ = 0, and for x = I, S = ^', hence 

F{ct) +f{ct) = Q and F,{ct + I) -f (ct-l) = 0, 

from which it follows that f = — F and f = — F^, and therefore 

f(ct-\-l)+f{ct-l) = 0, orf {ct,+%l)^ -f (cti). 

We see from the latter formula that the body, after the time ti = 

2 I 

— , will assume the opposite state of motion, and that it will con- 
c 

sequently make a complete vibration in double that time, 2 ti = 

— . The period of the complete vibration is, therefore, 
c 

'~ c ~^^ gE' 
or double that in the first two cases. 



1070 GENERAL PRINCIPLES OF MECHANICS. [§ 19. 

By means of these formulas we can calculate from the period t 
of a complete vibration or from the number n of vibrations, which 
a prismatical body makes in a given time, the modulus of elasticity 

jE' = (— ) . -, and the velocity of propagation or the velocity of 

sound in it, c = — . 
It 

Example. — An iron wire, which was 60 feet long and was fixed at both 
ends, was put in longitudinal vibration by means of friction in the direc- 
tion of its axis. The number of complete vibrations was 1637 in a second ; 
what was the modulus of elasticity of the wire and what was the velocity 
of propagation in it ? According to one of the above formulas, we have 
for the modulus of elasticity, expressed in units of length, 

L = ^- m^ i (2 n Vf = 'ilV^l' = 99870000 inches, 
g \ t / g ' 33,2 . 12 ' 

and if a cubic inch of this iron weighs 0,28 pound, the modulus of elasti- 
city, expressed in pounds, is 

E = 99870000 . 0,28 = 27960000 pounds (compare the table, § 212). 

The velocity of propagation, or the velocity of sound in it, is 



c = ^/g L = V32,2 . 99870000 .^ = ^ 16,1 . 16645000 = 16370 feet, 
or, assuming the velocity of sound in the air to be c = 1092 feet, we have 

_ 16370 _ 

"" - t6m: - ^^' 

If the vibrating wire is very long, the period of a vibration depends 

upon the length of the wave or upon the distance I between two nodes, 

2? 
audit is always t^ = — . This time determines t\iQ pitch of the note pro- 

c 

duced by the vibrating wire ; the greater or smaller t^ is, the lower or 

higher the note is. The intensity of the sound, on the contrary, increases 

with the amplitude of the vibration. For spherical waves, in which sound 

propagates itself in air and water, c and t remain unchanged, and it is only 

the amplitude of the vibration, or the intensity of the sound, which 

diminishes. 

(§ 19.) Transverse Vibrations of a String. — The transverse 
vibrations of a string or elastic rod can be treated in the same 
manner as the longitudinal ones. As the simplest case is that of a 
stretched string (Fr. corde ; Ger. Saite), we will discuss that first. 
Let A D B, Fig. 892, be any position of the vibrating string, A 
and B the two fixed points, I — A B the length of the string, G its 
weight and S the tension, which is to be regarded as constant. 
Now if ^ ]V= X and NO = u he the co-ordinates of any point of 



19.] 



THE THEORY OF OSCILLATION 



1071 



the string, and if we resolve the tension S at it into two components 
X and F, one parallel to A B, and the other perpendicular to it. 







Pi 


Fig. 89! 

s 


I. 






oj 


* 


.^^^^j 






'K y 




■ 




T 


U Ki 


^^^■^v^s. 




^ 


fir 


^N^ 


i^ 


\f 


--,_ 












V _ 


,,.-•*'' 
•••••• 



B 



we can regard the latter as the motive force a one end of the 
element Q. If the arc A = s \^ increased by the element 
Q = d s, and if the corresponding increase of the ordinate y is 
Q T = d y, P, S,d y and d s are the homologous sides of two sim- 
ilar triangles OPS and Q T 0, and we can put 

8 ~ OQ ~ ds'^ ds'^' 



But another force Pj 



R U 



S = 



dy, 



S, which is one of the 



Q E' ^^ ds 

components of the opposite tension, acts in the opposite direction 
upon the same element Q ; hence the motive force, which moves 
the element Q back to the axis A B, is 



P _ P, = i^-l^) 8. 



The mass M of this element is proportional to its length 
Q = d s; now if we suppose the amplitude y of the oscillation 
to be small, we can assume that the mass is proportional to the 

dx G 



element T — Q U = d x oi the abscissa, or that M — 



I ■ 9 



If we make this assumption, we have the acceleration with which 
the element approaches its position of rest A B 

dy, g SI 



- P - P^ _ ^y_ 

^ ~ M ~~ d s . d X 



G 



or, if we put d s =^ d x, 



_ dy - dy, g SI 

^ ~ dx' ' G ' 

d y 
Now y is some function of x, e.g. \\) {x) ; hence -— is another 

Cv JO 



function V>i {x) and 



d y — d y^ _ d d y _ ^ [i/>i {x) ] . 



d x" dx" 

function ip^2 (^) of this quantity, and 



d X 



IS a third 



1072 



GENERAL PRINCIPLES OF MECHANICS. 



[§3C. 



p = ip, {x) . 



G ' 



Since y is also a function of the time t, i.E. y — (t> {t)^ the Te- 
locity with which the element Q returns to its position of rest is 

V = -~- = (f)i (^), and the corresponding acceleration is 



dcp.it) 



02 (0- 



If we equate these two values of p, we obtain, as in 
Appendix, the differential equation 



17 of the 



</>2 (0 = V^2 {^) 



a 



= C' 1/^2 (^), 



and we can put here, as we did there, 

y = (f) (t) = ip (x) — F (c t + x) i- f (ct — x) and 

q; =c[Fi (c t + X) + /a {C t — x)]. 

Since here also for x = and x = I, y and ?; = 0, we have 
again/ ==■ — i^and f (c t + I) = f {c t — I), or/ (c ti -{- 2 I) = 
f {c ti) ; hence the period of a complete vibration or double oscil- 
lation is 

2 1 ^ 

t^ = —- ~ 2 I y zTTPif ^^y if we put G = A lyy 



f, = 21 



9 SI 

^ g8' 



The period of vibration of a string is therefore directly propor- 
tional to the length I and to the square root of the weight of the unit 
of length, and it is inversely proportional to the square root of the 
tension S of the string. ■ 

Example. — Since half the period of the vibration corresponds to that 
of the next octave, a string will give, according to tliis formula, the octave 
of the fundamental tone, when it is shortened one-lialf or supported in the 
middle, or when it is stretched four times as much, or when it is replaced 
by another whose unit of length weighs one-fourth as much as that of the 
first one. 



(§20 



Fig. 893. 



,) Transverse Vibrations of a Rod.— ^The period of vi- 
bration of an elastic'rod or spring 
A B (Fr. lame ; Ger. Stab), Fig. 
893, which is fixed at one end, 
can be determined in the follow- 
ing somewhat circuitous man- 
ner. According to § 226, if r 
denotes the radius of curvature 
of the rod at a certain point 0, 




§30.] . THE THEORY OF OSCILLATION. 1073 

determined by the co-ordinates C N —x^ and N = yx, the moment 
of flexure of the arc A = 5i is 

M = . 

r 

If we put the force, with which an element Q, which corre- 
sponds to the co-ordinates C R = x and E Q = y, approaches the 
axis or position of rest C B, = P d x, or its moment 

= N R . P dx =z [xx — x) P d x,we obtain 

= / (Xi — x) P d X. 

r t/o 

{xi — x) P d X = / P Xxdx — I P xdx 
— iCi / P dx ~ I P xdx, 
©r, if we put / P dx = Pi, and therefore 

I P X d X — / P dx . x = P^Xi ~ I PxdXy 

/ {xx — x) P dx — / P^dx'y hence we haye also 

WE /'"I „ ^ 
= / Pxdx. 

d s^ 

Now we know that r = — , „ ,.. r (see Art. 33 of the In- 

dx d {tang, a) ^ 

troduction to the Calculus), or, since we can put, when the deflec- 
tion is small, d s — d x, 

d X , 

r = 7-77 r; hence 

d (tang, a) 

^ -r,d {tana, a) /'■i „ 7 

a X J 

by differentiating which, we obtain 

_WE.di^-^^^^) = P,dx. 

TO 1 / / X ^ (^y , / N d (tana, a) 

If we put 2/ = V^ {x), tang, a = -^ = ip, {x), -^^ — '- 

= -02 {x) and d I — ^ f' — -j = 1/^3 (a;), we obtain the equation 

P,= - WE.ij,{x), 
by differentiating which again, we find 

dPx= - WEdi^s {x), I.E. P dx- - WEdipi {x), or 

P= _ WE^-^!^= - WE^,{x). 

(ti X 

68 



1074 GENERAL PRINCIPLES OF MECHANICS. . [%2h 

In order that tlie spring shall yihrate symmetrically, we can as- 
sume that P is proportional to y, or that P = — K y\ hence we 
have 

WEijj^ {x) = Ky, or ^4 (^) = -f^- «/ = ^'V^ 

when we denote tft^ by k^. 

This differential equation 1/^4 {x) — Ic'^y corresponds to the equa- 
tion y = \l) {x) = A COS. {h x) + B sin. {k x) + C e^"" + D e~~^''; 
for by successive differentiations we obtain 

ijj^ (x) = h [— A sin. {k x) + B cos. {k x) + C &"" — D e-^% 
xp, (x) = F [— A COS. (hx) ~ B sin. (k x) + G e'''' + D e"'*], 
1/^3 {x) = F [A sin. (k x) — B cos. {k x) + C e'"'' — D e~^ ''J, and 
1/;^ {x) — ¥ \^A COS. \kx) ■\- B sin. {k x) + C e^^ + D e^'^'"], 
so that we have really 

-04 (^) = F y, 

(§ 21.) The period of vibration t of the elastic rod is found, as 

force 

above, by substituting p = 02 (t) = . But the force acting 

mass 

upon an element is 

= Pdx=— Kydx= — WE¥ydx, 
and, when the cross-section is F and the heaviness is y, the mass is 

=^F d X — : hence 

9 
^ ,,, g W E¥ 

^2 (0 = Y^ -y = - l^ yy 

X. A ^ ^x. '9 WEk' , 
when we denote the expression - — ^ by fji . 

This differential equation corresponds to the simple formula 
y = (f) (t) = sin. {fi t -\- t), 
in which r expresses any arbitrary time of beginning ; for by dif- 
ferentiation we obtain 

d i/ 

V — -—r =: (f)^ (t) = jJi . cos. ([X t + r) and 
a V 

d V 
p — -^ = <^^{f) — — \i^ , sin. {}it -\- r), I.E., 

02 it) ^ - l^"" y- 
If in the equation y = sin. (fi t + r) we take r = 0, we obtain 
y = sin. (fjb t) ; hence for jU ^ = 0, tt, 2 tt, etc., y = 0, and conse- 
quently 



^31.] THE THEORY OF OSCILLATION. 1075 

^1 = — is the period of a simple vibration and 

, %n ^^ J Fy . ^, . , , , , . 

i^ = — - = -p- y ^^vfE "^^ period of a complete vibration. 

In order to calculate the period of a vibration, we must know 
not only the quantity k, but also the ratio -™. 

If the rod is cylindrical and its radius = r, we have 

F Ti r^ 4 
TIF — 1 T = — (see § 231), 

and if it is a parallelopipedon, whose width is 5 and whose height 
is y^, F 111 VI . ^ „^„. 

If = yr^ = i^ (^^' § ^^^)- 

We have, therefore, for the first rod 

< = ---/" 



and for the second 



^ = ^/ 



4Tr,/3r 



}i¥ ^ g E' 

The quantity Ic is found in the following manner from the 
equation 

y — A COS. {h X) + B sin. {h x) + C e^"" -\- D e'^"" . 
'If we substitute in this formula the corresponding values x == I 
and y =z 0, we obtain 

1) = ^ COS. (hi) + B sin. (k I) + C e'^' -h D e"*'. 
If we perform the same operation in the equation 

ta7ig. a = —^ =z ip^ (a:), we obtain 

2) = - ^ sin. (hi) + B cos. (k I) + G e'^' + D e-'^K 
Since the moment of flexure at the end A of the rod = and 

consequently the radius of curvature r = oo , or V^a {^) == and 

<^3 (x) = 0, it follows that 

= - ^ COS. - B sin. Q + C e' + D e-\ i.e., -^+(7+Z) = 

and 

= ^ sin. 0- B COS. +Ce' - D e-\ i.e., -B+0-D = 0, 

whence 3) A = C + D and 

4:) B = a-I). 
If we eliminate A and B from these four equations, we have 
{C + D) COS. {k I) ■\-{C- D) sin. {k I) +Ce^' + D e"*' = 0, and 
-{C + D) sin. {kl) + iC- D) cos. (k I) +Ce^' - D e-^' = ; 



1076 GENERAL PRINCIPLES OF MECHANICS. [§ 5Si. 

from which we obtain by addition 

Ccos.{kl) - Dsin.{kl) + Ge^^=0, 
and by subtraction 

D COS. {hi) + C sin, {h I) + D e-^\= 0, or 

C [cos. {Tc I) -]- e^^'] — D sin. {h I) and 

D [cos. (kl) -}- e-^^= — C sin. (k I) ; 

hence we have by division 

* cos. (h I) + e^^ sin. (h T) , 

— ^ — 7^-^ = 77-,-v — —Tr, whence 

sm. {/c I) cos. (fc I) + e'"^ 

2 + cos. {Tc T) {e^^ + e-^ = ^> ^"^ 
COS. {hl) = - -^n-z^-^v 

The smallest of the different values, which correspond to the 
different tones that the rod can give out and which depend upon 
the number of nodes, is ^ Z = 1,8751 ; the greater are, on the 
contrary, nearly i. ■> _ ^ ^ ^tt 7 tt 

If we are required to find from the observed period t of the 
complete vibration the modulus of elasticity B, we have generally 
to consider but the smallest value ; we must, therefore, put 

' Ic- i'^^^^ and F - ^^^-^ ■ 

hence for a cylindrical rod 

g\r ¥ tf g \3,516 rtl '9 r' f 

and for a parallelopipedical one 

^gXJik'' tl ~ ^g \d,blQht) ' ' g h'f 
Remark 1. — ^If we compare with each other the formulas 

t = —J y -^ and t, = 2l^ i/-^ 

for the transverse and longitudinal vibrations of one and the same rod, we 

obtain the proportion 

P 3,516 P 

tit.=— : -^ — L, I.E., t:t.=-~: 0,5596 l^. 

Wertheim found by experiment that this proportion was correct for 
cast steel and brass. 

Remakk 3. — The transverse vibrations of an elastic rod are discussed 
by Seebeck in a " Abhandlung der Leipziger Oesellschaft der Wissenschaften^^^ 
Leipzig, 1849, and also in the " Programme der technischen Bildungsan- 
stalt in Dresden," for the year 1846. Wertheim's experiments upon the 
elasticity of the metals and of wood by means of transverse and longitu- 



§22.] 



THE THEORY OF OSCILLATION. 



1077 



dinal vibrations are discussed at length in " Poggendorff's Annalen," 
Erganzungsband II, 1845. 

Remark 3. — The period of vibration or rather*the number of vibrations 
of a rod in a given time cannot generally be determined directly on 
account of their rapidity ; we must, therefore, employ various artifices to 
do it. We can determine it either, as Chladni, Savart, etc., did, by the 
pitch of the note produced by the vibration, or we can employ the method 
first proposed by Duhamel, which consists in causing the rod to describe 
by means of a small point a wave-line upon a revolving glass plate, which 
is covered with lamp-black. A chronometric appa/ratus^ to which a flying 
pinion^ such as used in the striking works of town clocks, is attached, is 
employed to produce a regular motion of rotation. An account of this 
apparatus is to be found in Morin's " Description des appareils dynamo- 
metriques, etc., Paris, 1838," as well as in his " Notions fondamentales de 
m^canique." Wertheim determined the number of vibrations in a given, 
time by allowing another body, such as a tuning-fork, whose number of 
vibrations was known, to vibrate at the same time with the rod to be ex- 
amined. If we cause both bodies to trace wave-lines upon the lamp-black 
and then count the number of waves corresponding to the same central 
angle, the ratio of these numbers will give the ratio of the numbers of 
vibrations. The longitudinal vibrations are generally accompanied by 
small transverse ones ; the rod describes, therefore, a corrugated wave- 
line. By counting the small waves contained in one large wave of the 
main wave-line, we can easily compare the number of longitudinal vibra- 
tions with the number of transverse ones. 

§ 22. Resistance to Vibration. — The forces, which cause the 
vibrations of a body, are very often accompanied hj passive resist^ 
ances, whose influence must be examined more particularly. If 
such a resistance is constant, as, e.g., the friction of a pendulum 
upon its axis or that of a magnetic needle upon its pivot, it has no 
influence upon the period of the oscillations, but their amplitude 
is diminished at every stroke. For the case in § 1 (Appendix), in 
which the motive force is proportional to the distance x from the 
position of rest or centre C of the motion A B, Fig. 894, we can put 

p = fj,x = fi(a — Xi), 

in which Xi denotes the space A M de- 
scribed. If we take into consideration 
the diminution h of this space, in con- 
sequence of the friction, we have, when 
the body is describing the first half 
A C of its path, 

p —\i{a —h — x^^ 




1078 GENERAL PRINCIPLES OF MECHANICS. [§23. 

and when it is describing the second half C B 

jp — — \i\_x^ — {a ^ 'k)\\ 
the influence of the friction h consists, therefore, in this alone, 
that for one-half of the path a must be replaced by a — /^ and for 
the other by ci + h, and that the whole space described in one 
oscillation must be changed from 2 « to 2 c^ — 2 ^, i.e. the ampli- 
tude of the oscillation will be diminished a certain quantity 2 Ic at 
each oscillation. Finally, since the amplitude does not enter into 
the formula + — '^ 

V ■ 



Ic can have no influence upon the period of the oscillations. 

The case is different with the resistance of the air. The latter, 
when the velocities, as in the case of the pendulum, are small, is 
more nearly proportional to the simple velocity than to its square, 
as was shown by Bessel's researches upon the length of the simple 
pendulum (Abhandl. der Akademie der Wissensch. zu Berlin, 1826). 
This is explained by the fact that this resistance is increased prin- 
cipally by the condensation and rarefaction of the air in front and 
behind the vibrating body, which increase with the velocity v of 
the body (see § 510 and Appendix, § 17, Eemark). In accordance 
with this assumption, we can put the acceleration of the vibrating 

body 

p^=^ — {lix + vv)oYp-\-vv-\-iix = 0, 

when we assume the body to be moving from the point of repose 
and measure the space from that point. 
If we put 

^=f {t)y '^ = ^ =.A (0 andi^ =: ^ =/2 {t), 

we can write also/g {t) + ^/i {t) + fJ^fit) = 0, which corresponds 
to the integral equation 

X = [i COS. {i> t VJi) + ^1 sin. ('0 t V^)'\ e~Y, 

in which b and hi denote constants to be determined and i/j = 

/ 1^ 

y 1 — -T-. ]S"ow for ^ = 0, a; = 0, whence J = ; hence we have 

more simply 

/- "A 
X = 1)1 sin. (ip t Va) e 2 . 

Since this value becomes = 0, when ip t Vjul = tt, the period of 
an oscillation or simple vibration is 



§23.] 



THE THEORY OF OSCILLATION. 



1079 



t=: 



77 



n 



i>Vfj, 



—J I-E. -7 = 



times as great as if the resistance of tlie air were not present. 

Remark. — It is easy to explain why bodies which are set iu vibration 
make smaller and smaller oscillations and finally come to rest. This efifect 
is due to two causes, the resistance of the air and the imperfect elasticity 
of the vibrating body ; in consequence of the latter fact, the contraction 
and expansion of the body, particularly within a short space of time, is not 
proportional to the forces acting upon it. 

§ 23. Oscillation of Water. — The simplest case of the wai'o 
motion of ivater is that presented by its oscillations in tivo communi- 
cating tubes A BCD, Fig. 895. Let us assume that both Lave 

the same cross-section, and let us imagine 
the surface of the water in one leg to be 
raised a certain distance H A = x above the 
position it occupies when at rest, and that in 
the other leg to be depressed an equal dis- 
tance R D = X. We have here the motive 

force 

P = A.'^xy, 

and if / denotes the entire length A B C D 



Fig. 89o. 




= H B C R of the water, the mass moved is M 



Aly, 



g 



; hence the 



acceleration with which the surface of the water rises or falls is 



P = 



P 2Axy 



9 



2 g X 



M Aly ^ I 

Since this formula corresponds exactly to the law of oscillation 
p = fj, X, discussed in § 1 and § 2 of the Appendix, we have for the 
period of an oscillation 

t = -—= = ny -—. 
VJi ^g 

Since the period of the oscillations of the simple pendulum, whose 
length is ^, is 

the oscillations of the water in the communicating tubes are iso- 
chronous with those of this pendulum. 

If both legs of the tube A B C D, Fig. 896, are inclined, i.e. if 



-080 



GENERAL PRINCIPLES OF MECHANICS. 



[§23. 



the axis of one of the tubes forms an angle a and that of the other 
an angle (i with the horizon, the space A ff = D R = x, which 
the surface of the water describes upwards in one and downwards 
in the other leg, corresponds to the difference of level 

z = X sin. a -\- X sin. (3 — x {si?i. a + sin. 13) • 
hence the force is 
Fia. 896. F = A y X {sin. a + sin. (3), 

the acceleration is 

ff {sin. a + sin. (3) .x - 




P 



I 



and the period of the oscillations is 

t = n\/- 



l 



g {sin. a + sin. (3) ' 

If, finally, the tubes are of different icidtlis, the determination of 
the period of the oscillations becomes much more complicated. Let 
A be the cross-section and I the length of the middle tube, a^, Ai 
and /i the angle of inclination, the cross-section and the length of 
one lateral tube, and a^, ^2 and ?2 the angle of inclination, the cross- 
section and the length of the other; finally, let us suppose that the 
surface of the water in the axis of one tube has risen a distance x 
and that the surface of the water in the axis of the other has sunk 
a distance Xc^. We have then 

Ai Xi = Aq X2, whence x^ = -^ x, 

and the motive force, reduced to A^, 

P — Ax {x-i^ sin. ax + iCg sin. aj y = —^- 



(^2 sin. Oj + ^1 sin. a^ x^. 



The mass of the water in the middle tube is constant and equal 



to 

A, 



AJy 
9 



, and, since the ratio of its velocity to that of the force is 



-^y the mass reduced to the point of application is 



(^).Aly. 



The mass of the water in the first leg is 



AAJn_±Jo,)y 
9 



, and that in the second 



__ ^2 (?2 — Xi) y 
~ 9 



S34.1 



THE THEORY OF OSCILLATION. 



1081 



or reduced to the point of application of the force 

— /^y ^(?2 - X,) y 



Finally the mass moved by P is 



ff 






-^•IL 



AiXi 



A.. 



) 



V VI Co •*/! 

f- — H -\ 

g \A Ai A, A, 



9 



rl + l 

U ^ A, 



A.. 



and the acceleration is 



P 



(sin. «! . sin. a^ 
-a;- + -A--) ^ "^ 



A ^ A,^ A, 



W" Z?)^^ 



Xr 



If the cross-sections of the two tubes were the same, we would 
have Ai = A. 2, and therefore 

{sin. a^ 4- sin. ae) ffXi__ {sin. a^ + sin. a^) g Xi 



P = 



(x * ^1-') ^. 






+ /l + ?2 



and the period of the oscillations 



^ ^ (5m aj + sin. a.2)' 

Remabk. — In consequence of the friction and of the resistance due to 
the bend in the tube, these formulas must, of course, be modified (com- 
pare Appendix, § 25). 

§ 24. Elliptical Oscillations. — If a body, which is driven with 

an acceleration p = iiz = ii.CM towards a fixed point C, Fig. 897, 

possesses an initial velocity c, whose 
direction differs from that of the 
force, the oscillations no longer take 
place in a straight line, but in an 
elhpse, as we will now proceed to 
prove. Let the direction of the mo- 
tion at the point of beginning A be 
at right angles to the distance C A 
= fl^ and let the corresponding ve- 
locity be = c. If we pass the co-ordi- 
nate axes through 0, one upon and the other at right angles to 




1082 GENERAL PRINCIPLES OF MECHANICS. [§24. 

G A, and denote the co-ordinates C K and K M h^ x and y, we 
have for the components q and r oi p := ii z, which are parallel to 

the axes, since - = - and - = -, 

p z p z 

q = jji X and r = fji y. 

If u and V are the components of the velocity w of the body Jf, 
which are parallel to the axis, we have, according to § 1 of the 
Appendix, 

ti = Vfi {a" — x")', 
and at the same time 

c^ — v" = z I Tdy=^%\ijydy — \iy-^ whence v = \^& — \i ?/^ 

Since for y = Z*, v — 0, it follows that 

= (3^ — jii ^^ ; hence c — h \^\i and v — \^ii {h' — y"^). 

cl X d u 

But now u = ^-7 and v ~ —r, and therefore 

dt dt 



u d X . /a^ — x^ d X d ii 

- = ^- — V T2 -2 0^ ,- - ,- "^ - , I.E., 

V dy ' V -f Va: - x' Vh' - f 



'0 ^(1) 



hence (according to Art. 26, V, of the Introduction to the Calculus) 

sinr^ - = sinr^ - + Con. 
a a 

or, since for a; = «,?/— 0, 

. i« . , ™ 

8%n. - = s%n. Y + don., or 
a 

sinr^ 1 = sinr'^ + 6^o^., i.e., — == (7o?^. and 

s^^. - =r sm. Y + ttj or 
« 5 2 

sm. stn. ^ = — -. 

^ h % 

When the dijfference of two arcs is — , the sine of one is equal 
to the cosine of the other, i.e., 



= ^^Hf . «' ©■- (!)■= '• 



§24.] THE THEORY OF OSCILLATION. 1083 

Since this is the equation of an ellipse, it follows that a point, 
which is impelled or attracted towards C with an acceleration ii z, 
will describe an ellipse, whose semi-axes are 6^ ^ = a and C B — h. 

We have also 

d t = —^— , ^ ; hence the time is 

t — y -smr^ ^, or inversely, 

y = h sin. (t VJt) and x = a cos. (t VJi). 
The time, in which the body will describe a quadrant of the ellipse, 
is found by putting y z= t^ and it is 

, i/l . _i ^ i/l . _i 1 ^ 

ti = y - sm. ' -- = y - sm. 1 = - — — . 

The time, in which the body describes half the ellipse, is 



and the period of a complete revolution or of a complete vibration is 

or exactly the same as it would be, if the motion were a rectilinear 
reciprocating one. It follows also that 

u — Vfi («' — a:') = ^[j, (a"" — a"" [cos. {t ^/^)]') = fJ' a sin. {t Vfi) 
and 

V — Vfi {h'' — y') = fib COS. {t Vfi) ; 
hence the velocity of revolution is 



to = Vu' -i- v' = fi ^{a sin. t ^\if 4- {b cos. t Vfi)\ 
Finally, we can put 

a ■]- h /, ./-x . a — h 



X = 



COS. (t Vy) -\ — cos. {t Vfji) and 



^ V^ ■ r-. . 2 



y — — ^ — sin. (t Vfji) — sin. {t Vfx) ; 

now since the first members 

— ^ — COS. (t Vfi) and — ^ — sin. {t Vjl) 

rt J- Jk 

correspond to a uniform motion in a circle, whose radius is — ^ — , 
and since the two other members correspond to an opposite uni- 



1084 



GENERAJL PRINCIPLES OF MECHANICS. 



[§25. 



form motion in a circle, whose radius is 



a 



2 



-, we can also assume 



that the elliptical motion of the point is composed of two circular 
ones, I.E., that the point describes uniformly a circle, whose radius 

, while the centre of the latter moves uniformly in a circle. 



IS 



2 



whose radius is 



a + d 



If J = 0, the oscillation takes place in a straight line, out we 
can imagine it to be composed of two equal opposite circular 
motions. 

§ 25. Waves of Water. — According to the accurate obser- 
vations of the Weber brothers, an example of elliptical oscillatiq^i 
is presented by the motion of waves of water (Fr. ondes ; Ger. 
Wasserwellen). Not only every particle on the surface, but also 
every particle below it describes in the wave motion an ellipse. 
On account of the resistance on the bottom the ellipses below the sur- 
face of the water are smaller than those at it, and in general they de- 
crease with the distance from that surface. The different elements 
in the surface of the water, as well as those in any other plane 
parallel to it, are at the same moment in different phases of mo- 
tion ; while an element A, Fig. 898, is beginning its path at (0), 

Fig. 898. 




an element B is already at (1), a second C is at (2), a .third D 
at (3), a fourth E at (4) ; at this moment the vertical section 
of the surface of the water is a cydoidal or troclioidal curve 
A BCDEFGHJ. Before the wave motion began, the ele- 
ments were at the centres K, L . , . N oi their trajectories and 
formed the horizontal surface K JSf oi the water ; during the wave 
motion, on the contrary, part of the elements are above and part 
are below this line, and all have, of course, a tendency to return to 



%25.] 



THE THEORY OF OSCILLATION. 



1085 



their positions of rest K, L , . , N. The oscillations are, however, 
elliptical so long only as the waves remain unchanged ; if they de- 
crease gradually in magnitude, the path of each element becomes 
narrower and narrower and no longer forms an ellipse, but a spiral 
line. On the other hand, when the waves are forming or increas- 
ing in size, the elliptical trajectory is formed gradually from a 
spiral line. 

After one instant A has moved in its trajectory to (1), B to (2), 
(7 to (3), etc., and the wave-form has been moved forward in conse- 
quence through the horizontal distance K L between two elements ; 
after a second instant A is at (2), B is at (3), O is at (4), and the 
wave-form has again advanced the distance K L ^ L M\ thus, as 
the elements of the water revolve, the wave-form advances more 
and more, and when an element has made a complete revolution, 
the wave has advanced its own length K N. When an element has 
made half a revolution, as is shown in Fig. 899, the place of the 

Fig. 899. 




tuave-crest is occupied by a trough or sinus, and that of the latter 
by a crest. This advance of the wave-form does not, of course, 
consist in any particular motion of the water, but in the forward 
motion of the same phase, e.g., in the forward motion of the crest 
J" (Fig. 898) of the wave to 0, P, etc. If the period of a revolu- 
tion t of an element of the water and the length ^ / = s of a wave 
are known, we can calculate the velocity of propagatio7i by means of 
g 

the formula '^ = r- 
f 

The height of a wave, or the sum of the height of the crest and 

the depth of the trough is equal to the vertical axis 2 J of the 

ellipse, in which the elements of the water revolve ; the length C G 

of the trough exceeds the half length of the wave by the length 2 a 

of the horizontal axis of the ellipse, and the length of the crest is, of 



1086 GENERAL PRINCIPLES OF MECHANICS. [§36. 

course, that much shorter than half the wave length. Hence the 
cross-section of the trough of a wave is larger than that of the wave- 
crest ; now since this is impossible in consequence of the invariabil- 
ity of the volume of the water, the centre of the elliptical trajectory 
must be somewhat above the surface of the water when it is at rest. 

§ 26. Webers' Experiments. — According to Webers' experi- 
ments, the path described by a particle of the water at the surface 
of a wave is a slightly compressed ellipse ; according to Emy, on 
the contrary, the particles of water in sea-waves describe upright 
ellipses. Both axes of the elliptical path decrease as the depth 
below the surface increases, and according to Weber the horizontal 
axis decreases more rapidly than the vertical one. The wave ap- 
pears not to be propagated in a vertical direction ; elements verti- 
cally below each other are, according to the observations of the 
Weber brothers, in the same phase at the same time ; on the con- 
trary, those situated in a horizontal line form a complete series of 
the different phases of the motion. From the experiments cited 
above, it appears that the period of revolution of an element, or the 
time in which a wave is propagated its own length, depends prin- 
cipally upon the ratio of the two axes of the path. The greater 
the ratio of the horizontal axes 2 a to the vertical one 2 b, the 
greater is the period of revolution. The particles, which lie deeper, 
describe their paths more quickly than those at the surface ; from 
this we must conclude that the wave length diminishes towards 
the bottom. 

The velocity of propagation c = - of a wave depends, since the 

Ob 

time of revolution t increases with the ratio t> not only upon the 

length 5, but also upon the height h. If a wave is propagated be- 
tween two parallel walls, e.g. in a canal, its width remains con- 
stant, its height h diminishes and its length increases in such a 
manner that the only change in the velocity of propagation is that 
resulting from the friction of the water upon the walls. If, on the 
contrary, a wave can propagate itself freely in all directions, and if 
it forms a wall which recedes into itself, its length and width are 
both increased at the expense of its height, and the wave becomes 
gradually flatter and flatter until in a short time the eye is no longer 
able to distinguish it. If such a wave is not originally circular it 
will gradually approach the circular form as it advances. Accord- 
ing to Webers' experiments, the height diminishes in arithmetical 



P7.] THE THEORY OP OSCILLATION. 1087 

progression when the wave advances in geometrical progression. 
The velocity of propagation of such a wave diminishes gradually, 
the farther the wave is propagated. If, on the contrary, a wave is 
propagated from without inwards and is contracted more and more 
in consequence, its height, length and velocity gradually increase. 
There is, therefore, a great difference between the waves of water 
and those of sound. In the latter the velocity of propagation de- 
pends upon the elasticity and density of the medium alone ; in the 
former, on the contrary, it is a function of the length and height. 
If the undulations of the water are produced by a force which acts 
almost instantaneously, e.g., by the immersion and quick with- 
drawal of a solid body, the particles of the water describe elliptical 
paths which gradually decrease, or rather spiral lines, which draw 
themselves together more and more, and the periods of revolution 
become smaller and smaller. The origin of a whole series of waves, 
which become smaller and smaller, is to be attributed to these rela- 
tions of motion. As the waves are propagated farther and farther, 
those which follow are increased in size by those which have pre- 
ceded them, and those most in advance in a short time become so 
flat as to be invisible. This running together of the waves gives 
rise to systems of small waves, which present themselves like teeth 
upon the front surface of the main wave. These small waves or 
teeth advance, according to Poisson and Cauchy, with uniformly 
accelerated motion. 

§ 27. Hagen's Experiments. — According to the latest in- 
vestigations of Geli. Oherhaicrath Hagen (see the " Seeufer-und 
Hafenbau von G. Hagen, Berlin, 1863," 1 Vol., which forms the 
third part of that author's " Wasserbaukunst ;" also his treatise 
upon waves in water of uniform depth ; Berlin, 1862), the particles 
of water of waves in deep water describe with constant angular 
velocity circles, whose diameters decrease as the depth increases, and 
at the bottom they are infinitely small. A filament of water, which 
when at rest is vertical, will oscillate, in consequence of the wave 
motion, backwards and forwards about this vertical line, its base 
remaining fixed very much as a stalk of wheat is moved by the 
wind. The line of the wave or the curve which unites the points, 
which are in the same phase of revolution and which, when the 
water is at rest, is a straight line, is therefore a prolate cycloid, 
that becomes more and more prolate as the depth increases ; at 
the bottom it is nearly a straight line and at the surface it ap- 



1088 GENERAL PRINCIPLES OF MECHANICS. [§28. 

proaches the common cycloid. From the radius f of the common 
cycloid, whose value for high sea- waves rises to 50 feet, we obtain 
the length of the wave Z = 2 rr r, its velocity of propagation 

c = \/%Jr - |/^-, 
the period of a wave 

c ' g ^ g 

and the angular velocity with which the molecules of water describe 

Q 

their elliptical paths, w =r -. 

The centre of the circle, in which a particle which is situated 
lower down revolves, is determined from the radius z of this circle 
and from its distance y from the centre of the first circle, whose 
radius is r, by means of the formula 



y 



= <l)- 



By inversion we obtain z = i- e~~ , in which e = 2,71828 de- 
notes the base of the Naperian system of logarithms. We can 
easily understand from this that the circles of oscillation decrease 
very rapidly with the depth ; for r = 10 feet, at the depth y = bO 
feet, ^ = 10 . e-°'^ = 3,50 feet, and at the depth y = 200 feet, 
:= 10 . e-"''' = 0,15 feet. 

When the waves are of small constant depth, as Mr. Scott 
Eussel had already remarked, the horizontal motions of the parti- 
cles of water, which lie above one another, are equally great ; the 
filament of water, which was originally vertical, remains so during 
the wave motion, but its length and thickness vary. The different 
particles describe closed curves of equal horizontal diameters and 
of variable vertical ones, which decreases gradually with the depth ; 
they are, however, ellipses only when we suppose that the height 
of the wave is infinitely small compared to the depth of the water. 

When the depth of the water is finite and the height of the 
waves is great, the laws of the motion of the waves are very com- 
plicated. 

§ 28. Interference of "Waves of Water. — If two water- 
waves cross each other, the same general phenomena occur as in 
the case of waves of air and other fluids ; after they cross each other, 
each wave continues its motion as if they had not met ; but accord- 



28.] 



THE THEORY OF OSCILLATION. 



1089 



ing to Weber's observations, it is accompanied by a small loss of time, 
so that a wave requires a little more time to pass from one point to 
another when it passes through another wave than when it is prop- 
agated freely. If two crests come together, a crest twice as high as 
the lirst is produced, and in like manner when two troughs meet, a 
third, twice as deep, is formed. According to Weber's experiments, 
the ratio of the height of the simple wave to that of the compound 
one is 1 : 1,79. When two waves interfere, or when a wave-crest 
coincides with a trough of a wave, the two counterbalance each 
other, and the point where this occurs remains at the same level as 
the surface of the still water. The paths of the single particles, 
when two waves meet, become straight lines, which are vertical at 
the crest, but at a distance from it their positions are such that 
they are inclined towards the crest. 

K a wave of water imjnnges against a solid wall, it will be re- 
flected by it as if it came from a point as far behind the wall as 
that from which the wave started is in front of it, and the reflected 
wave will pass through the one which is arriving exactly in the 
same manner as any two waves, which cross each other, do. 

In Fig. 900, 1, II to V, the phenomena, which are presented 

Fig. 900. 




when a wave A B C D E is reflected by a rigid wall M N, are re- 
presented. In I the crest C D £J of sl wave is arriving at the wall 



GO 



1090 



GENERAL PRINCIPLES OF MECHANICS. 



[§28. 



M N and the reflection begins in the form of a wave Ci Dx E^ ; in 

II the top of the crest D of the wave has arrived at the wall and 
has combined with the half D^ E^ of the reflected crest of the wave ; 
half a crest C G oi almost double the height is thus produced. In 

III the trough A B C of the wave has just reached the wall, while 
the reflected crest Ci Dx Ex is passing over it ; an interference is 
thus produced which causes the wave to disappear entirely. In IV 
the bottom B of the trough of the approaching wave coincides witli 
the bottom Bx of the trough of the reflected wave \ a trough A S 
of double the depth is thus formed. Finally, in V the approaching 
wave A B C I) E is reflected completely by the wall M N and thus 
changed into the wave Ai Bi C\ Dx Ex, which moves in the oppo- 
site direction. 

Fig. 901. 




When the waves are reflected by a wall, the paths of the mole- 
cules undergo the same changes ^s when two waves cross each 
other; here also, in the neighborhood of the wall, the horizontal 
component of this motion is more and more balanced, and, on the 
contrary, the vertical one is increased more and more, so that near 
the wall the path becomes a vertical line, and farther from it an 
inclined one. If the wave strikes obliquely against the wall, it will 
be reflected, like every elastic body, at the same angle at which it 
struck. If a wave strikes but partially against an obstacle, the 



28.] 



THE THEORY OF OSCILLATIONS 



1091 



phenomena of inflexion are produced, new waves being formed at 
the extreme ends of the obstacle. 

Finally, stationary luaves of water, like those of a string or any 
other solid body, are formed when two waves of the same length, 
which originate at two points situated at a distance apart equal to 
1, 3, 5, 7 . . . times the fourth part of the length of a wave, cross 
each other. Ij^t A B C D E F G H, Fig. 902, 1 and II, be one, and 
A.BiCi D, E, F, Gx H, the other wave. At the points K, L, M, N, 
where the two systems of waves are at the same distance from, but 
on opposite sides of the middle line, the motions counteract each 
other and fixed points of interference are produced; on the con- 
trary, above and below the points 0, P, Q, R, where the two wave- 
lines cut each other and the paths are therefore doubled, the tops 
of the crests and the bottoms of the troughs are alternately formed. 

Fig. 902. 




Kemakk.— The most complete treatise upon the motion of waves is the 
following : " Wellenlehre auf Experimente gegriindet. etc.," by the brothers 
G. H. Weber and W. Weber, Leipzig, 1825. A good abstract of it is con- 
tained in the " Lehrbuch der Mechanischen Naturlehre," by August. Mid- 
ler's " Lehrbuch der Physik und Meteorologie," Vol. I, can also be con- 
sulted. The treatises of Laplace, Lagrange, Flaugergues, Gerstner and 
Poisson are reviewed and criticised in Weber's work. Cauchy's " Wellen- 
Theorie" and Bidone's "Versuche" are discussed at length in "Gehler's 
Physikalisches Worterbuch," Art. " Wellen." Emy's wave theory has been 
translated by Wiesenfeld and published under the title " Ueber die Be- 
wegung der Wellen uiid liber den Bau am Meere und im Meere," Vienna, 
1839. Hngen's work lias silready been cited, § 27. The theory of water- 
waves has been treated by Airy in an article upon " Tides and Waves," in 
the Encyclopadia Metropolitan a. 



TRANSLATOR'S APPENDIX. 



O INCE the last German edition of the present volume was issued 
the author has published in the " Civihngenieur'' several articles 
upon subjects, which have been treated in the foregoing pages. 
As they contain much valuable information and give the results 
of a very great number of very careful experiments, a brief abstract 
of the matter contained in some of them will be given here. Those 
which will first be noticed are three articles upon the efi&ux of 
water, viz. : 

(1) the different methods of experimenting upon the efflux of 
water under a constant head (Die verschiedenen Methoden der 
Versuche uber den Ausfluss des Wassers unter constantem Drucke. 

X Band, 1 Heft) ; 

(2) experiments upon the efflux of water under a very small 
head (Versuche iiber den Ausfluss des Wassers unter sehr kleinem 
Drucke. X Band, 3 und 4 Heft) ; 

(3) the relations of compound efflux, considered theoretically 
and illustrated by experiment (Die zusammengesetzten Ausfluss- 
verhaltnisse theoretisch entwickelt und durch Versuche erlautert. 

XI Band, 2 und 3 Heft). 

Article No. 1 begins with a description of the various methods 
adopted by different experimenters to maintain a constant head in 
the main or discharging reservoir. Smeaton returned the water, 
which was discharged, to the reservoir by a hand-pump and thus 
maintained the water level constant in the former. Christian em- 
ployed a large weighted cask, wiiich was suspended by a rope ; as 
the water was discharged from the reservoir, the cask was allowed 
to sink so as to displace exactly the same quantity of water as had. 
flowed out -of the reservoir. In Prony's experiments the escaping 
water was caught in a vessel, which was connected with two paral- 



TRANSLATOR'S APPENDIX. 1093 

lelopipedical cases (made of sheet-metal). The latter floated upon 
the water in the main reservoir, and the apparatus was so arranged 
that the increase in weight of the vessel caused the floats to dis- 
place exactly the same quantity of water as had been discharged'. 
The impulse of the escapmg water will interfere with the working 
of this apparatus, unless proper precautions are taken. Hachette 
(see his " Traite elementaire des Machines ") passed a hollow tube 
through the bottom of the reservoir ; by sliding the tube up or 
down the level of the water in the reservoir could be changed. If 
the volume of the water, which entered the reservoir, exceeded thft. 
discharge, the excess escaped over the top of the tube. A slight 
variation of level, of course, took place. The author tried several 
different methods of obtaining the same result. The first, which 
to a certain extent resembles Smeaton's, was to feed the discharg- 
ing reservoir from the main reservoir by means of a pipe, in which 
an ordinary cock was placed. An assistant is stationed at the cock, 
by turning which he maintains the surface of the water in the 
discharging reservoir at a constant level, which is marked by a 
fixed pointer in the reservoir. The second method he employed 
was Christian's. He used a hollow float made of sheet-metal ; its 
weight could be regulated by filling it partially with sand. By 
allowing the float to sink as the water was discharged, the surface 
of the water was maintained at a constant level, which was indi- 
cated by a pointer. The volume of the float gives the discharge. 
This method is not so accurate as that last described (by means of a 
cock), and it is not so simple as it appears at first sight ; for the 
size of the float must vary with that of the orifice. The floating 
sypJion gives more accurate results than Prony's apparatus, de- 
scribed above. It consists essentially of a T-shaped syphon with 
two lateral pipes, by which the water enters, and of a larger central 
pipe, by which it leaves the apparatus. Each of the lateral pipes 
passes through a water-tight cylinder of sheet-metal, which is open 
on top and floats upon the water. These two floating cylinders 
support the s}^hon ; by fllling them partially with water we can 
immerse the inlet orifices of the syphon as deep as we please, and 
the outlet orifice can be brought to any desired distance below the 
level of the surface of the water in the reservoir. As the surface 
of the water in the reservoir sinks, the whole apparatus descends 
with it, and the head or distance of the outlet orifice below the 
level of the water remains constant. 

The author has also applied the principle of Mariotte's flask to 



1094 GENERAL PRINCIPLES OF MECHANICS. 

maintaining a constant head, or constant velocity of efflux. The 
discharging reservoir is a cylindrical vessel, which is provided with 
two orifices or openings, but which is in all other respects air-tight 
One of these oj)enings is in the top and the other is upon the side 
near the bottom. A tube or pipe, w^hich is open at both ends, fits 
in the orifice in the top by means of an air-tight ground joint, in 
which it can slide up and down. The orifice in the side was so 
arranged that mouth-pieces of various kinds and sizes could be 
inserted in it. The vessel is first filled with water through the 
upper orifice and the pipe is then inserted and pushed down a cer- 
tain distance, depending upon the head we wish to have ; the ori- 
fice of efflux is then opened and the water in the tube sinks until 
air begins to pass under the bottom of the tube and rise to the top 
of the vessel. The head is now constant and is measured by the 
difference of level between the orifice of efflux and the bottom of 
the tube. In order to prevent the air, which enters through the 
tube, from causing too much disturbance, the bottom of the tube 
is surrounded by a cylinder of wire-gauze. A glass tube, which is 
open on top, enters the vessel at the bottom and is turned ver- 
tical upwards, serves to measure the pressure. The same principle 
can be applied in another form. An air-tight vessel, which is 
filled with water, has a pipe inserted in the side near the bottom ; 
this pipe passes below the level of the water in the discharging 
vessel. Another pipe, which is smaller and is made principally of 
India-rubber, enters the air-tight vessel near the top, and the other 
end of it is placed so as just to touch the surface of the water in the 
discharging reservoir. If the level of the water in the latter sinks, 
air enters the tube and water is discharged from the air-tight ves- 
sel, in consequence of which the surface of the water in the dis- 
charging reservoir rises and seals the mouth of India-rubber tube 
and the flow of water into the main reservoir ceases. The objec- 
tion to this method is the unsteadiness of the surface of the water, 
which renders it difficult to measure the head with accuracy. In 
order to render it more steady Geh. Oherbmiratli Hagen had two 
small holes made in the side of the large tube just above the outlet 
and in addition employed an intermediate vessel. 

A series of experiments, made with the aid of the different ap- 
paratus just described, gave the following results. The w^ater was 
discharged through an orifice in a thin plate 1 centimeter in 
diameter. 



TRANSLATOR'S APPENDIX. 



1095 



TABLE. 



Zs 
|'£ 

1 

2 
3 
4 
5 


Nature of the head. 


Description of the apparatus. 


Head in meters. 


Value of ^.' 

i 

i 

1 


Gradually decreasing. 
Constant 

a 
u 
u 

Average of tlie above 


Author's ordinary ap- 
paratus for experi- 
ments upon efflux . . 

Level maintained by 
a cock 

Level maintained by 
a floating body . . . 

Level maintained by 
Mariotte's flask . . . 

Level maintained by 
apparatus last de- 
scribed 

five experiments .... 


Ui^ =0,1700 
I Ag = 0,0500 

h = 0,100 
u 


1 

I 
1 

1 

0,6647 

i 

0,6776 
0,6576 
0,6518 

0,6654 
0,6634 





By the aid of one of the above-described apparatus, experiments 
upon efflux with constant influx can be made. The formula to be 
employed (see page 923) is 

t = --^-^ ( Vh - VT, + VI I {"^h-Vk \. 

The discharging reservoir which was used in these experiments 
was the apparatus represented upon page 927 ; by means of Mari- 
otte's flask, the discharge per second into the former was main- 
tained constant during each experiment. In these experiments 
the surface of the water in the discharging reservoir either rose or 
fell. By preliminary experiments, the coefficients of efflux for the 
orifices in both vessels were determined. 

In the first experiment the surface of the water in the discharg- 
ing reservoir rose. The observed duration of efflux was t — 170,25 
seconds ; that calculated by the above formula from the data given 
by the experiment was t = 170,5 seconds. 

In the second experiment the surface of the water sank ; the 
observed time was t = 213,2 seconds, the calculated was 213,9 
seconds. 

A.nother case, which often occurs in practice, is that represented 
in Fig. 776, page 908, when the reservoir A C is very large com- 
pared to G L. The water passes from the large reservoir A C 



1096 GENERAL PRINCIPLES OF MECHANICS. 

through a pipe, into the reservoir G L, from which it is discharged 
through the orifice F into the air. By prolonging the discharge 
pipe of Mariotte's flask so that it will reach below the surface of 
the water in the discharging reservoir (Fig. 792), the level of which 
surface is variable during the experiment, we obtain an example 
of this case. The formula for the duration of efflux, which must 
be employed, is 



t = 



[(/. FY + ill, FY] V2 g 



Vh + Vh Vx — vy 



^^^^lVT.-Vlc\ ^y 






V li\ + V'h^ V y 
in which G denotes the cross-section of the main discharging res- 
ervoir, F the area of the orifice in the main reservoir, /t its coeffi- 
cient of efflux, i^i the cross-section of the outlet orifice of Mariotte's 
flask, pLi its coefficient of efflux, h^ the height of the surface of the 
water in the main reservoir above the orifice in it, h the height of 
the constant water level in Mariotte's flask above the variable one 
in the main reservoir, x what h becomes in the time t, y what hi 
becomes in the time U, and h^ = h -^ hi = x •\- y ) Tc \^ the value 
of X, when the flow becomes permanent, i.e. 

_ {ill F,Y K 

(fiFY + {fi.Fr 

and 

A/j ^^ iIq — rC, 

In the first experiment the surface of the water in the main 
reservoir sank; the observed value of ^ was 116,33 seconds and the 
calculated value was 116,67 seconds. In the second experiment 
the level of the water rose ; the observed time was t — 157,5 seconds, 
and the calculated value of t was 158,18 seconds. 

No. (2.) Experiments upon the Efflux of Water under 
a very small head. — From previous experiments by the author 
and others, we know that for an orifice in a thin plate one centi- 
meter in diameter, 

1, when the head is 103,578 meters, fi = 0,600 

2, " " 13,574 " ju = 0,632 

3, « « 0,909 ." u = 0,641 

4, « " 0,101 " fi = 0,665, 



TRANSLATOR'S APPENDIX. 



1097 



2, 
3, 

5, 



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and that for a brass tube 1 centimeter in diameter and 2 meters 
long, the coefficient of resistance 

1, when the velocity is ?; = 20,99, is 4" = 0,01690 

V = 12,32, is C = 0,01784 

V = 8,64, is C = 0,01869 

V = 2,02, is ^ = 0,02725 

V = 0,57, IS ^ = 0,03646 ; 

but we have no experiments which show how the coefficient of 
efflux increases, when the head is very small (e.g. 1 to 2 centime* 
ters). It is also important to know how ^ increases, when the 
velocity of the water is very small (e.g. 0,1 meter). In the above- 
mentioned article the author gives a detailed description of a very 
extended series of experiments, undertaken for the purpose of dis- 
covering the above relations. The discharging reservoir was a 
wooden trough 2,25 meters long, 0,45 meters wide, and 0,190 me- 
ters deep. It was necessary to make the reservoir as long and wide 
as possible ; for the surface of the water could, of course, sink but 
a very short distance during the experiment. The author then 
gives a description of the various methods and apparatus employed 
to determine with accuracy the cross-section of the orifices and the 
head of water. This portion of the article, although of the greatest 
interest, would be out of place here. 

The table on page 1098 gives the results of the experiments 
with orifices in a thin plate and with other mouth -pieces. 

The temperature of the water was between 15° and 18° Centigrade. 

From the 8 experiments with orifices in a thin plate (No 1 to 
3^0. 5), whose diameters varied from 0,405 to 2,529 centimeters, we 
see that the contraction diminishes, when the head is small, as it 
does when the head is large, not only with the head, but also with 
the diameter of the orifice. 

From the data given in the table on page 1098 and at the be- 
ginning of the article, the following table has been arranged. 



Head h 


0,020 


0,101 


0,909 


13,574 


103,578 


CoeflBcient of efflux ^ 


0,711 


0,665 


0,641 


0,632 


0,600 



The experiments under Nos. 6 and 7 show that in this case also 
the coefficient of contraction for an orifice in a thin conically con- 
vergent wall is greater than that for an orifice of the same size in a 



1098 



GENERAL PRINCIPLES OF MECHANICS. 










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TRANSLATOR'S APPENDIX. 1099 

thin plate, and that it is less for an orifice in a conically divergent 
wall than for the latter. In experiment No. 18 a free contracted 
stream could not be obtained. The efflux took place with a filled 
tube and the stream pulsated quite violently. 

It was also observed that the discharge was not increased as 
much by rounding off the inlet orifices of the ajutages, when the 
head was small as when it was great. 

The table on page 1100 contains the results of experiments with 
long tubes made of glass, brass and zinc. Preliminary experiments 
were made to determine the coefficients of resistance of the inlet 
and outlet mouth-pieces combined. By subtracting the coefficients 
thus found from those obtained for the long tube and inlet and 
outlet mouth-pieces together, the author deduced the coefficient of 
resistance for the tube alone. 

These experiments showed the coefficient of resistance ^ of the 
water to be very great, when the velocity is small. This coeffi- 
cient ^ IS nearly the same for glass and bi'ass tubes. 

To the table 

for V = 20,99, ^ = 0,01690 
'' = 12,32, C = 0,01784 
" = 8,64, ^ = 0,01869 
" = 2,02, ^ = 0,02725 

« = 0,485, ; = 0,03453, 
we can now' add 

for V -.= 0,2028, ^ = 0,0587 
« = 0,0890, C = 0,1420. 

The third portion of the article is devoted to an account of a 
series of experiments upon the flow of water through bends and 
elbow^s under a very small head. The coefficient of resistance for 
the inlet and outlet portion was first determined as in the experi- 
ments, the results of which are given in the last table. The table 
on page 1101 contains the coefficients of resistance for the flow of 
water through elbows and bends under small heads. 

We see from the last table that the coefficients of resistance of 
elbows are much greater than those for bends of the same diameter, 
when both cause the direction of the motion of the water to change 
90° and when the radius of curvature of the axis of the bend is 
equal to the diameter of the tube. 

' The third article (No. 3), which is cited above, is very long, 
covering 68 columns of the Civilingenieur. As it would be impos- 
ible to condense the matter contained in it in the limited space 



1100 



GENERAL PRINCIPLES OP MECHANICS. 



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1102 GENERAL PRINCIPLES OF MECHANICS. 

which is at our disposal, we will content ourselyes with an enumer- 
ation of the subjects treated. They are — 

(1.) The simultaneous discharge of water through two orifices, 
when the head diminishes. 

(2.) The variable discharge of water from one vessel into a sec- 
ond, in which the orifice is submerged, while a constant quantity 
of water is continually discharged into the first vessel. 

(3.) The variable efllux of water through a notch, either with or 
without influx. 

(4.) Efflux of water from a prismatical vessel, with free influx 
into the latter from another prismatical vessel. 

(5.) Efflux of water from a prismatical vessel, with influx imder 
water from another prismatical reservoir. 

These cases are treated at length ; the formulas are first deduced 
and then tested by very careful experiments. Any one interested 
in the subject of hydraulics will find this article worthy of his 
most attentive perusal. 

We would also call attention to the following articles by the 
author upon subjects connected with hydraulics. 

"Hydrometric experiments upon the application of the formulas 
of Daniel Bernouilli (page 804) and Charles Borda (page 884), as 
well as upon the use of a new water-meter; also upon the friction 
of water in conical pipes and upon the play of jets d'eau" ("Hydro- 
metrische Versuche iiber die Anwendung der Formeln von Daniel 
Bernouilli und Charles Borda, so wie iiber den Gebrauch eines 
neuen Wassermessers (einer Wasseruhr) ; ferner fiber die Eeibung 
des Wassers in conischen Eohren und fiber das Spiel von springen^ 
den Wasserstrahlen," Civilingenieur, Band XIII, 1 Heft). " Com- 
parative hydrometric measurements by means of a tachometer, a 
large rectangular orifice of efflux and a large overfall extending 
across the whole wall," (" Yergleichende hydrometrische Messungen 
mittels eines hydrometrischen Elugelrades, einer grosseren rec- 
tangularen Ausflussmundung und eines grosseren fiber die ganze 
Wand weggehenden Uberfalls," Civilingenieur, Band XIII, 5 and 6 
Heft). 

The latter article contains an account of Schwamkrug's water- 
divider, mentioned upon page 986. 

I. " The quicksilver differential piezometer and its application 
to the determination of the difference of the pressure of the water 
in a set of conduit pipes." 



TRANSLATOR'S APPENDIX. 1103 

II. "The water piezometer with a micrometer, as well as its 
application to the determination of the pressure of gas in pipes, 
etc." 

III. " A supplement to the article cited above upon the differ- 
ent methods of experimenting upon efflux under a constant head." 
('•I. Das Quecksilber-Differentialpiezometer, etc. II. Das Wasser- 
piezometer mit Mikrometer, etc. III. Eine Ergiinzung der Ab- 
handlung iiber die verschiedenen Methoden der Ausflussversuche 
unter constantem Drucke." Civilingenieur, Band XV, 2 Heft.) 

The translator would also call attention to two articles by the 
author upon "experimental mechanics," which form apart of a yet 
unpublished work upon that subject. The titles of the articles are : 

(1.) " Experiments to accompany lectures upon the elasticity 
and strength of solid bodies " (" Versuche bei Vortragen iiber Elas- 
ticitat und Eestigkeit fester Korper," Civilingenieur, Band IX, 
5 Heft), and 

(2) "Experiments to accompany lectures upon Mechanics" 
("Versuche bei Vortragen iiber Mechanik," Civilingenieur, Band 
XIV, 6 Heft). 

The first article contains a description of the apparatus used 
by the author in experimenting before the students at Freiberg 
upon flexure and torsion. By means of this apparatus, which is 
very simple and easily constructed, the professor can show to the class 
almost all the phenomena of flexure and torsion. He can also de- 
termine the moduli of rupture and of elasticity not only by observ- 
ing the deflection and angle of torsion, but also by allowing the 
body to be experimented upon to vibrate and counting the number 
of vibrations. The modulus of resilience and that of fragility can 
also be determined. No. (2) contains an account of some modifi- 
cations of the above apparatus, by means of which experiments 
upon the theory of couples (including their composition and de- 
composition) can be made. This is followed by the description of 
a simple reversable pendulum, by means of which the value of fj 
can be determined in the lecture-room with little difficulty. The 
author then takes up the subject of the elasticity of rigid bodies. 
He discusses four cases of double flexure : first, that of a prismati- 
cal rod of a rectangular cross-section, bent by a force, whose direc- 
tion forms an angle 6 (which is not 90°) with one of the sides of 
the cross-section; secondly, that when the cross-section of the rod 
is a right-angled triangle and the direction of the force is perpen- 
dicular to the base of the triangle ; thirdly, that when the rod is 



1104 GENERAL PRINCIPLES OF MECHANICS. 

acted upon by two forces, whose lines of action do not lie in the 
same plane ; and fourthly, that when the beam is bent in the shape 
of an elbow and loaded at the extreme end with a weight (the 
crank is an example of this case). The article closes with an ac- 
count of some experiments with compound girders. 

Those engaged in teaching will find the last two articles full of 
valuable information ; but a translation of them would occupy too 
much space here. 

In conclusion, we would mention an article upon "the flexure 
of a homogeneous prismatical measuring rod, supported in two 
points, as well as the shortening of its length, produced by it, dis- 
cussed in as elementary a manner as possible " {" die Biegung eines 
in zwei Punkten unterstiitzten homogenen prismatischen Mess- 
stabes, sowie die durch dieselbe her vorgebrachte Ye rkiirzung seines 
Langenmaasses, auf moglichst einfache Weise ermittelt von Julius 
Weisbach," Civilingenieur, Band XII, 4 Heft), 



IN DEX. 



5%* figures give the number of the page* 



A. 



Aberration of the stars, 152. 
Abscissas, 34. 
Acceleration, 108, 113, 134. 

" along the abscissas, 146. 

« « u ordinates, 146. 

normal, 143, 607. 
of gra^^tJ, 113, 159. 
Adhesion, force of, 163, 763. 

plates, 762. 
Aerodynamics, aerostatics, 165. 
Aggregation, state of, 163. 
Air balloon, 798. 
' efflux of, 933, 934, 939. 
* heaviness of, 795. 
' layers of, 787. 
' manometer, 796. 
' pressure of the, 777. 
' pump, 790. 
Amplitude of an oscillation, 649, 1043. 
Angular acceleration, 576. 

velocity, 576. 
Antifriction pivots, 349. 
Aperture of efflux, 800. 
Apparatus for hydraulic experiments, 

936. 
Application, point of, 163, 193. 
Arc, length of an, 85. 
Archimedes, principle of, 757. 
Areometers, hydrometers, 758. 
Arithmetical mean, 97. 
Arm of the lever, 195. 
Ascension, vertical, 116. 
Asymptote, 49, 51, 53. 
Atmosphere, pressure of the atmo- 
sphere, 777, 787. 
Attraction, the law of magnetic, 1056. 
At wood's machine, 599. 
Axes, free, 634. 

" principal, 634. 
Axis, neutral, 410. 
" of a couple, 205. 



Axis of revolution or rotation, 906, 

348, 573, 639. 
Axis, pressure upon the, 350. 
Axles, friction on, 311, 316. 



B. 



Balance, hydrostatic, 756. 

torsion, 1050. 
Ballistic pendulum, 693. 
Barometer, 776. 

" measurement of heights 

with the, 788. 
Beam, 418, 423, 437, 430. 

" subjected to a tensile force, 559. 
Bed of a river, 955. 
Bending, flexure, 409. 

" rupture by, 453. 
Bends, curved pipes, 896. 
Bent lever, 356. 
Binomial function, 57. 

" series, 57. 
Bodies, material, 154. 

of uniform strength, 387, 498, 
504, 539. 
" rigid, flexible, elastic, 380. 
Boilers, thickness of, 738. 
Bottom of the channel, 955. 

" pressure on the, 781. 
Brachystochronism, 659. 
Brittle, 373. 

Buoyant effort, upward thrust, 742, 
797. 



c. 



Capillarity, 762. 

Capillary tubes, 772. 

Cataract, 876. 

Catenary, 293 ; common catenary, 299. 

Central impact, 667, 669. 

Centre of gravity, 213. 



1106 



INDEX. 



Centre of mass, 213, 574, 
" " oscillation, 661. 
" " parallel forces, 205. 
" " percussion, 637, 692. 
" " pressure of water, 725. 
Centres, 349. 
Centrifugal force, 608. 

" of water, 719, 720. 
" " work done by, 610. 

Centripetal force, 608. 
Chain bridge, 292. 

" friction, 358, 361. 
Cinematics, 154. 
Circle, 34. 

" centre of gravity of an arc of 

a, 216. 
" oscillatory, 87, 142, 415. 
Circular functions, 70. 
Cistern barometer, 776. 

" manometer, 779. 
Clack valves, 900, 905. 
Cloistered arch, 243. 
Cocks, 900, 903. 
Cohesion, 371, 762. 

force of, 163. 
Collar bearings, 347. 
Columns, proof load of, 532. 
Combined elasticity and strength, 

373, 547. 
Communicating pipes, 723, 761. 
Components, 129, 174, 177, 1071. 
Component velocities, 129. 
Composed forces, 174. 

" motions, 126. 

Composition and decomposition of ve- 
locities and accelerations, 131, 132. 
Composition and decomposition of 

forces, 174, 177, 179, 195, 207. 
Composition and decomposition of 

couples, 202. 
Compound discharging vessels, 907. 

" pendulum, 661. 

Compressed air, work done by, 783, 

936. 
Compression and extension, 374. 

" elastic and permanent, 

376. 
strength of, 372, 373. 
Concavity, 39, 55. 
Conduit pipes, 874. 
Conical pivots, 347. 

" tubes or pipes, 873. 
" valves, 905. 
Connecting rod, 537, 573. 
Constant factors, 41, 61. 
force, 166. 
" members, 41, 61. 
" quantities, 33, 41. 
Contracted vein or stream of water, 
821, 823. 



Contraction, coeflScient of contraction, 

822, 944. 
Contraction, complete and incomplete 

or partial, 837. 
Contraction, perfect and imperfect, 840, 

858, 887. 
Contraction, scale of, 836. 
Convexity, 39, 55. 
Coordinates, 34. 

oblique, 79. 
Cosine and cotangent, functions of, 71. 
Couple, 200, 412. 

" axis of a, 205. 
Crank, 121. 
Cross-section, 376, 676, 801, 955. 

." weak, dangerous, 495. 

" sudden variation of, 883. 

Curvature, radius oi', 87, 142, 413. 
Curve, elastic, 414, 417. 
Curved surfaces, 40. 
Curves, convex, concave, 39, 44, 54 
" quadrature of, 78. 
" rectification of, 85. 
Curvilinear motion, 141, 145, 189. 
Cycloid, cycloidal pendulum, 655, 656. 
Cylinder, hollow, 443. 



D. 



Dam, 732. 

Daniel Bernouilli, 804. 

Decomposition and composition of 

couples, 202. 
Decomposition and composition of 

forces, 174, 177, 179, 195, 207. 
Decomposition and composition of 

velocities and accelerations, 131, 133. 
Density of bodies (specific gravity), 161. 
Dependent variable, 33. 
Deviation, angle of, 895. 
Differential, 38. 

" ratio or quotient, 39. 

Directive force of the magnetic needle, 

1053. 
Discharge, 800, 933. 
Discharge-pipe of a dam, 858, 922. 
Displacement, angle of displacement, 

530, 649. 
Diving-bell, 783. 
Ductility, 372. 
Dynamics, 155, 165. 



E. 



Earth, magnetism of the, 1054, 1059. 
Efflux, coefficient of, for water, 834. 
" " air, 944. 
" from moving vessels, 817. 



INDEX. 



1107 



Efflux of air from vessels, 932, 934. 
939 941. 
" of different fluids, 805, 930. 
" of moviug water, 843. 
" of water under water, 806. 
" of water from vessels, 800. 
" under variable pressure, 910, 

952. 
" velocity of, 800. 
" with filled tube, 853. • 
Elastic curve, 4l4, 417, 522. 
" extension, 375, 404. 
" fluids, 712. 
Elasticity, 163. 371. 1045. 

limit of, 371, 376. 
modulus of, 378, 407, 1049. 
Elbows, 894. 
Elevation, angle of, 136. 
Ellipse, 50, 284. 
Ellipsoid, 594. 
Elliptical oscillation, 1081. 
Emptying of a vessel, 910. 
Energy, 168. 

" of discharging water, 801. 
Envelope, 139. 
Equality of forces, 156. 
Equilibrium, 155. 

kinds of, 249, 250, 264. 
indifferent, 250. 266. 
Evolute, 88. 

Expansive force of steam, 35. 
Expansion by heat, 793. 

coefficient of, 793 
of the air, 781 
Exponential function, 63. 
Extension, elastic and permanent, 375, 
394. 
" experiments upon, 393. 

F. 

Fall of a stream, 955. 

" of bodies, 35, 113, 639, 659. 
Filling and emptying locks, 924. 
Final velocity, 108. 
Flexure, 409. 

strength of, 373, 450. 
moment of, 412, 414, 432, 436. 
Flotation, axis of, plane of, 746. 
Floating, depth of floatation, 745, 749, 
756. 
" bodies, floating spheres, 989. 
staff, 990. 
Fluids, 162, 712. 
Force, direction of a, 163. 
Force, living, 171, 173. 
Forces, measure of, 158. 
" moment of, 195, 414 
" normal, 143, 607. 
tensile, 374. 



Forces, 154, 155, 163,205. 

" equality of, 156. 
Fragility, modulus of, 383, 453. 
Free axes, 624. 
Freshet or flood, 973. 
Friction, resistance of friction, 309. 
angle of, 314. 
" balance, 317. 

coeflicient of, 313. 
" coefficient of, of air in pipes, 

949. 
*' coeflficient of, of water in pipes, 
864. 
. " coefficient of, of water in riv- 
ers, 965. 
cone of, 314. 
" height of resistance of, 864. 
kinds of, 310. 
laws of 311. 
of axles, 311, 316. 
" rolling, 353. 
" upon inclined plane, 323. 
wheels, 336. 
work done by, 313, 335. 
Fulcrum, 256. 
Function (;»"), 44. 
Functions, 33. 
i'unicular machine, 280. 
polygon, 286. 



G. 



Gases, aeriform bodies, 776. 

Gas-meters, 1023. 

Gauging, 976. 

Gay-Lussac's law, 793. 

Geostatics, geodynamics, geomechan- 

ics, 165. 
Girder, 418, 422, 427, 430, 464. 

" hollow and webbed, 437, 477. 
Goblet, hydrometric, 986. 
Gram, kilogram, 157. 
Graphic rej^resentation, 34, 122. 
Gravity, 113, 154, 163. 

'' centre of, 213. 

" determination of the centre of, 
214. 

" plane of, line of gravity, 213. 

" specific, 161. 
Gudgeons, 311. 
Guldinus, properties of, 241. 
Gyration, radius of, 581, 608. 



H. 

Hard, 372. 
Hardness, 676. 

Head of water, height of water, 722 
801, 809. 



1108 



INDEX. 



Heat, force of, 163. 
Heat, work done by, 936. 
Heaviness, 180. 

" mean, of the earth, 1051. 
of air, 795. 
" steam, 795. 
" water, 160. 
Height dae to the velocity, 115, 809. 

of rise, height of fall, 116, 878. 
Horizontal and vertical pressure, 732, 

736, 742. 
Hydraulic observatory, 995. 
Hydraulics, 165. 

Hydrometers, Hydrometry, 976, 989. 
Hydrometric goblet, 986. 

" pendulum, 999. 

Hydrostatic balance, 757. 
Hydrostatics, hydrodynamics, 165. 
Hyperbola, 51, 80. 



I. 



Impact, different kinds of, 667, 668. 
direct, 667. 
duration of, 668. 
elastic, 668. 
friction of, 685. 
imperfectly elastic, 680. 
line of impact, 667 
oblique, 668, 682. 
strength of, 702, 705. 
Impulse, 1002, 1006. 

" of air or wind, 1030. 

" water, 1006, 1011, 1029. 
Incidence, angle of, 6S4. 
Inclination, angle of, 314, 639. 
Inch, water, 983. 
Inclined plane, 272, 274, 639. 
Inertia, 157. 

force of, 157, 163, 574. 
" moment of, 576. 
Inflexion, 1091. 

point of, 55, 424. 
Integral, integral calculus, 60. 

" formulas, 73. 
Integration by parts, 76. 
Intensity of a force, 164. 

" the earth's magnetism, 

1060. 
Interference of waves, 1064, 1089. 
Interpolation, 98. 
Isochronism, 640, 658, 659. 



J. 



Jets of water, 876. 

Journals, trunnions, gudgeons, axles, 
305, 311, 345. 



Rater's pendulum, 665. 

Kilogram, 157. 

Knee lever, 257. 

Knife edges and points, 853. 

Knots, 281. 



L. 



Law of Gay-Lussac, 793. 

" " Mariotte, 37, 780. 
Laws of nature, 35. 
Length of a wave, 1064, 1085. 
Lesbros' experiments, 846. 
Lever, arm of, 195. 

" bent, 257. 

" kinds of, 255, 256, 343. 
Limit of elasticity, 371, 376. 
Line of current, mid-channel, 956. 

" " gravity, 213. 

" " impact, 667. 

" " rest, 743. 

" " support, 743. 
Load, proof, 379. 

" eccentric, 480. 
Locks, 924. 
Logarithm, 64. 
Longitudinal vibration, 1045. 
Loss of mechanical effect in impact, 

674, 883. 

M. 

MacLaurin's series, 57. 
Magnetic force, 163, 1056. 

needle, 1053. 
Magnetism, 1054, 1059. 

of the earth, 1054. 
Malleability, 872. 
Manometer, 776, 718. 
Mariotte's law, 37, 780. 
Mass, 158. 

" moment of, 577. 
Material pendulum, 661. 

" point, 165. 
Matter, 156. 
Maximum and minimum, 53. 

" " contraction, 

834. 
" " tension, 515. 

Mean, arithmetical, 97. 

" harmonic, 675. 
Mechanical effect, 168, 187, 209. 

" " loss of, during im- 

pact, 674, 883. 
" " of compressed air, 

783, 936. 
" of friction, 313, 335. 



INDEX. 



1109 



Meclianical effect of heat, 936. 

•' of inertia, 171, 577. 
" " of the centrifugal 

force, 612. 
Mercury, efflux of, 930. 
Metacentre, 751. 
Metal springs, 506. 
Method of least squares, 95. 
" " interpolation, 98. 
Mid-channel, line of current, 956. 
Modulus of elasticity, 378, 407, 1049. 
" " logarithms, 65. 

" proof strength, 380, 457, 
529. 
" " resilience and fragility, 

383, 453. 
" " rupture, or of ultimate 
strength, 380, 452. 
VIolecular action, 762. 
Molecules, molecular forces, 163, 762. 
Moment, magnetic, 1054, 1060. 
of a couple, 200, 201. 
" inertia, 577. 
" parallel forces, 207. 
" statical, 195. 
Momentum of a body, 670. 
Motion, absolute and relative, 1C5, 149. 
'• accelerated, retarded, 106. 
curvilinear, 141, 145, 189. 
" in resisting media, 1035. 
*' kinds of, 573. 
" of air in pipes, 950. 
" " water in channels, 955, 969. 
" " water in pipes, 869. 
" translation, 573. 
phases of, 1062. 
" rectilinear and curvilinear, 

105. 
" simple and composed, 126. 
" uniform and variable, 106. 



N. 

Naperian logarithms, 64, 80. 
Natural philosophy, 154. 
Nature, laws of, 35. 
Neil's parabola, 86. 
Neutral axis, surface. 410. 
Nicholson's hydrometer, 759. 
Normal, 87. 

acceleration. 143, 607. 

force, 189, 607. 
Notches, overfalls, weirs, 811, 914. 
Numbers, natural series of, 59. 



0. 

Obelisk, efflux from an, 919. 



Obelisk, centre of gravity of, 234. 
Oblique coordinates, 79. 
Observatory, hydraulic, 995. 
Oil, efflux of, 930. 
Ordinates, 34. 

" acceleration along the, 146. 

" velocity along the, 145. 

Orifices in a thin I'late, 821, 930, 944. 
inlet and outlet, 875, 880. 
of efflux, 800. 
" rectangular, 812, 828, 842, 846. 
Oscillation, 649, 1042. 

amplitude of an, 649, 1043. 
centre of, 661. 
period of an, 649, 1043, 1067. 
" of a pendulum, 649. 

" of the magnetic needle, 

1055. 
of water, 1079. 
Overfalls, notches, weirs, 811, 833, 844, 
849, 914. 



P. 



Parabola, 3, 87, 133, 291, 302. 
Parabolic motion, 134, 141. 
Paraboloid, 591, 720. 
Parallel forces, 199. 
plates, 770. 
Parallelogram of accelerations, 132. 
" forces, 17';. 
" " motions, 127. 

" velocities, 128. 
Parallelopipedon of velocities, 132. 
Pendulum, ballistic, 693. 
bob of a,- 591. 
" compound, 649, 661. 

" hydrometric, 999. 

Kater's, 665. 
" oscillation of a, 649. 

" reversable, 665. 

*' rocking, 665. 

" simple, mathematical, 648, 

661. 
Perfect fluids, 712. 
Percussion, centre of. 637, 692. 

point of, 692. 
Period, periodic motion, 106, 121. 
Permanency, state of, of running wa» 

ter, 957. 
Permanent extension or set. 375, 394. 
Phoronomics, 105, 154. 

formulas of, 119. 
Piezometer, 779, 881. 
Pile driving, 698. 
Pipes, long, 863. 

" thickness of. 738. 
Piston rod, 538, 573. 
Pitot's tube, 998. 
Pivots, friction of, 345o 



1110 



INDEX. 



Plane, inclined, 272, 323. 
'■ of revolution, 248. 
Pneumatics, 165. 
Point of application, 163, 192. 
" " inflexion, 54. 
" " suspension, 249, 664. 
Polyhedron, centre of gravity of, 231. 
Poncelet's orifice of efflux, 828. 

" tlieorem, 341. 

Position, 105, 150. 

" relative, relative motion, 150. 
Pound, 157. 

Powers, natural series of, 64. 
Pressure, hydraulic, hydrodynamic, 
808. 
hydrostatic, 713, 723, 724. 
' in water, 724. 
of the atmosphere, 777, 787. 
on the bcttom, 721. 
" vertical, horizontal, 732. 

Principal axes, 624. 
Principle of equal pressure, 713. 
Profile, longitudinal and transverse, 
955. 
" transverse, of running water, 
955. 
Projectile, path of a, 1038. 
Projectiles, height attained by, range 
of, 136. 
" motion of, in the air, 136. 

" motion of, in vacuo, 1038. 

Prony's method of measuring water, 

982. 
Proof load, proof strength, 379. 451. 

" moment of, 451, 472. 
Proof strength, modulus of, 380, 457, 

529. 
Propagation, velocity of, 1062, 1085. 
Properties of Guldinus, 241. 
Prosaphy and synaphy, 763. 
Pull, traction, 156, 374. 
Pulley, fixed and movable, 303, 304, 

368, 601. 
Puppet valve, 905. 



a 



Quadrature of curves, 78. 
Quantities, constant and variable, 33. 
Quicksilver, efflu-x of, 930. 
Quotient %, 93. 

" diflTerential of a, 43. 



Radius of curvature, 87, 14^, 413. 

" gyration, 581, 609. 
Ram, 698. 



Reaction, 164. 

of effluent water, 1002. 
wheel, 1015. 
Rectification of curves, 85. 
Reduction of a force, 255. 
" " masses, 578. 

" " the moment of flexure, 

432. 
" " the moment of inertia, 

580. 
Reflection, angle of, 684. 
Regulating apparatus, 900. 
Representation, graphic, 34, 122. 
Resilience, modiilus of, 383, 453. 
Resistance, coefficient of, 856, 884. 
height of, 856. 
of water, 1028. 
" to buckling or breaking 

across, 535. 
" to compression, 376, 392. 

Resistances, 155, 309. 

" passive, 1077. 

Rest, absolute, relative, 105. 
Resultant, 174, 177, 194. 
Revolution, axis of, 205, 248, 573, 629. 
plane of, 248. 
" solids and surfaces of, 238, 

241, 242, 693, 626. 
Rheometer, 1001. 

Rigidity of cordage and chains, 361, 363. 
" of hemp and wire ropes, 364, 
366. 
River, bed of a, 955. 
Rocking, rocking pendulum, 665. 
Rod, vibration of a, 1072. 
Rolling down an inclined plane, 646. 
friction, 353. 
of bodies, 605. 
Rotary motion, 210, 211. 
Rotation, axis of, 205, 248, 573, 629. 
plane of, 248. 
time of, 609. 
Running water, 955. 
Rupture by breaking across, 535. 
modulus of, 381, 452. 
" plane of, cross-section of, 495- 



S. 



Scale of velocities of a stream, 957. 
Set, permanent extension, 375, 394 
Sheering force, 412, 510. 

strength of, 373, 406. 
Shoots, efflux through, 848, 850. 
Short pipes, conical, 861, 891. 

" " conical convergent, 861. 

" " conical divergent, 862. 

" « cylindrical. 853, 888. 

« " efflux through, 852, 854 



INDEX. 



1111 



Short pipes, inclined, 857. 
" " interior, 855. 

Simpson's rule, 81. 
Sine, curve of, 7l. 

" fuTiction of the, 70. 
Sliding, 310, 639. 

" down an inclined plane when 
friction is considered, 643. 
Slope of a stream, 955. 
Soft, 372. 

Sound, velocity of, 1066. 
Sounding rod, sounding chain, 991. 
Specific gravity, 161, 755. 
Sphere, 227, 236, 588, 605, 646, 747, 918. 
Spheroid, 237, 588. 
Springs, spring dynamometer, 506. 

force of, 163. 
Statics, 155, 165. 
Stability, 250, 264, 269. 

" of floating bodies, 750. 
Steam, expansive force of, 35. 

" heaviness of, 795. 
Steel springs, 506. 

" tempered and annealed, 403. 
Stereometer, 788. 
Straight line, 49. 
Strength, 372. 

" of buckling or breaking' 
across, 535. 
ultimate, 379, 380. 
String, vibrations of a stretched, 1070. 
Subnormal, 87. 
Subtangent, 40, 66, 292. 
Surface, neutral, 410. 

of water, 719. 
Surfaces, curved, 40. 
Symmetrical bodies, 215. 
Symmetry, axis of, plane of, 215. 
Syphon manometer, 778. 



T. 



Tachometer, Woltmann's, 992. 
Tangent, tangential angle, 39, 47, 146. 
" function of, curve of, 71. 
" plane, 40. 
Tangential acceleration, 144. 
force, 189. 
velocity, 146. 
Tantochronism, 659. 
Temperature, 793. 
Tension, 281. 775, 776, 793. 

horizontal and vertical, 287. 
Theorem, Poncelet's, 341. 
Thickness of boilers and pipes, 738. 
Throttle-valve, 901, 903 
Top, 610. 
Torsion, 372, 523. 

angle of, 534. 



Torsion balance, 1050. 

elasticity of, 373, 523. 
" moment of, 524. 
" pendulum, vibrations due to 
torsion, 1050. 
strength of, 373, 528. 
Traction, pull, 156, 374. 
Tractrix, 850. 

Translation, motion of, 573. 
Transverse vibrations, 1048, 1070. 

" profile of running water, 

955, 959. 
Trigonetrical functions, 70. 

lines, 72. 
Twisting couple, 564. 
Tubes, conical, convergent, 861. 
divergent. 862. 
" short, efflux through, 852, 854. 
" conical, 861, 891. 
" cylindrical, 853, 888. 
" inclined, 557. 
" " interior, 855. 

" long or pipes, 863. 



u. 



Ultimate strength, modulus of, 880, 

452. 
Unguents, 310. 
Uniform motion, 106. 
Uniformly accelerated, uniformly re- 
tarded motion, 107, 108, 
112. 
" varied motion, 107. 

Unit of weight, 157. 

" " work, 169. 
Upward thrust, buoyant effort, 743, 
797. 



V. 



Valve-gate, 900, 903. 
Valves, 776, 779, 904. 
clack, 900, 905. 
puppet, 905. 
throttle, 901, 903. 
Variable, variable quantity, 33. 
" dependent, 33. 
" independent, 33. 
motion, 106, 117. 
" " of running watei; 

969 
Velocity, 107. 

" along the abscissas, 146. 
" along the ordinates, 146. 
coefficient of, 824, 944. 
final, 108. 
*' height due to the, 115, 809. 
initial, 108. 



1112 



INDEX 



Velocity, mean , 121, 124, 956. 

of propagation, 1062, 1085. 

of running water, 956. 

of sound, 1066. 

sudden variation of, 885. 

virtual, 187, 209, 212, 275. 
Vibi'ation of a stretched string, 1070. 

of an elastic rod, 1072. 
Virtual velocity, 185, 209, 212, 275. 
Vis viva, principle of, 171, 174 
Volume, 156. 
Volumeter, 789. 



w. 

Water, apparatus for measuring, 976. 
efflux of, 800. 
heaviness of, 160. 
height of in communicating 

tubes, 723, 761. 
hydraulic pressure of, 808. 
hydrostatic pressure of, 722. 
inch, 983. 
jets of, 138. 
meters. 1020. 
running. 955. 
stream of, 801, 821. 
surface of, 718, 765, 767. 



Water, waves of, 1084. 
Waves, 1062. 

" crest and trough of, 1085. 
height of, length of, 1085. 
of water, 1084. 
Web, 478, 479. 
Wedge, 277, 329, 496. 
Weight, absolute, 156, 159, 161. 

unit of, 157. 
Weir, overfall, notch, 811, 833, 844, 

849, 914. 
Wheel and axle, 305, 567, 595. 
Work done by a force, mechanical ef- 
fect, 168, 187, 209. 
" " friction, 313, 335. 
" " heat, 936. 
" " inertia, 171, 577. 
" unit of, 169. 
Working load, 380. 



X. 



Ximenes' experiments on friction, 318. 
" water vane, 1001. 



z. 



ZoB«. 593. 



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ALEXANDER, J. H.— Universal Dictionary of "Weights and Mea- 
sures, Ancient and Modern, reduced to the Standards of the United 
States of America. 

New edition, enlarged. Svo, cloth 3 50 

ANDERSON, Gen. ROBERT.— Evolutions of Field Batteries of Ar- 
tillery. 

Translated from the French, and arranged for the Army and Militia of 
the United States. Published by order of the War Department. 83 
plates. 24mo, cloth , 1 00 

ANDREWS, Maj. -Gen. C. C— Campaign of Mobile. 

Including the Co-operation of General Wilson's Cavalry in Alabama. 
Witli five maps and views. Svo, cloth 2 50 

r Hints to Company Officers on their Military Duties. 

ISmo, cloth „ 50 

ARNOLD, Maj. A. K.— Cavalry Service. 

Notes on Horses for Cavalry Service, embodying the Quality, Purchase, 
Care, and Diseases most frequently encountered, with Lessons for Bitting 

the Horse and Bending the Neck. Illustrated. 18mo, cloth 75 

1 



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D. VAN NOSTRAND'S CATALOGUE. 



ARNOLD, Maj. FRANK S— The Discipline and Drill of Militia. 
Crown 8vo, limp cloth $2 00 

ATWOOD, Geo.— Practical Blow-Pipe Assaying. 

12mo, cloth, illustrated 2 00 

AUGHINCLOSS, W. S.— Link and Valve Motions Simplified. 

Illustrated wi,th 37 wood-cuts and 21 lithographic plates, together with 
a Travel Scale and numerous useful tables. 8vo, cloth 3 00 

AXON, W. E. A.— The Mechanic's Friend. 

A Collection of JJleceipts and Practical Suggestions Eelating to Aquaria — 
Bronzmg— Cements — Drawing — Dyes — Electricity — Gilding— Glass-work- 
ing — Glues — Horology — Lacquers — Locomotives — Magnetism — Metal- 
working — Modelling — Photography — Pyrotechny — Railways — Solders — 
Steam-Engine — Telegraphy — Taxidermy -- Varnishes — Waterproofing, 
and Miscellaneous Tools, Instruments, Machines, and Processes connected 
with the Chemical and Mechanic Arts. With numerous diagrams and 
wood-cuts. Fancy cloth ... 1 50 

BACON, F. W.— A Treatise on the Richards Steam-Engine In- 
dicator, with directions for its use. 

By Charles T. Porter. Revised, with notes and large additions as de- 
veloped by American practice ; with an appendix containing useful for- 
mulae and rules for engineers. Illustrated. Fourth edition. 12mo, 
cloth 100 

BARB A, J.— The Use of Steel for Constructive Purposes ; 

Method of Working, Applying, and Testing Plates and Brass. With a 
Preface by A. L. Holley, C.E. 12rao, cloth 1 50 

BARNARD, Maj.-Gen. J. G.— The "C. S. A." and the Battle of 
Bull Run. 

8vo, cloth 1 25 

The Peninsular Campaign and its Antecedents, 

As developed by the Report of Maj.-Gen. Geo. B. MeClellan and other 

published Documents. 8vo, cloth 1 00 

12mo, paper 30 

Notes on Sea-Coast Defence. 

Consisting of Sea-Coast Fortification; the Fifteen-Inch Gun; and Case- 
mate Embrasure. With an engraved plate of the Fifteen-Inch Gun. 
8vo, cloth 2 00 

BARNARD, Maj.-Gen. J. G., and BARRY, Maj.-Gen. W. F.— Re- 
port of the Engineer and Artillery Operations of the Army 
of the Potomac, 

From its Organization to the Close of the Peninsular Campaign. Illus- 
trated by 18 maps, plans, etc. 8vo, cloth 2 50 

BARNES, Lieut.-Com. JOHN S.— Submarine Warfare, Defensive 
and Offensive. 

Comprising a full and complete History of the invention of the Torpedo, 
its employment in War, and results of its use. Descriptions of the va- 
rious forms of Torpedoes, Submarine Batteries, and Torpedo Boats actual- 
ly used in War. With 20 lithographic plates and many wood-cuts. 8vo, 
cloth 5 00 



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